lectures/lecture2/lecture2.ipynb

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    "from traitlets.config.manager import BaseJSONConfigManager\n",
    "import jupyter_core\n",
    "# path = \"/Users/i.oseledets/anaconda2/envs/teaching/etc/jupyter/nbconfig\"\n",
    "cm = BaseJSONConfigManager(config_dir=path)\n",
    "cm.update(\"livereveal\", {\n",
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    "              \"start_slideshow_at\": \"selected\",\n",
    "               \"scroll\": True\n",
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   "source": [
    "# Лекция 2. Ранг матрицы, малоранговая аппроксимация, SVD. Линейные системы"
   ]
  },
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   "source": [
    "## На предыдущей лекции\n",
    "\n",
    "- Форматы представления действительных чисел\n",
    "- Устойчивость алгоритмов вычисления\n",
    "- Векторы и векторные нормы\n",
    "- Матричные нормы\n",
    "- Унитарные матрицы: матрицы Гивенса и Хаусхолдера"
   ]
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   "source": [
    "## Сегодняшняя лекция (первая часть)\n",
    "\n",
    "- Ранг матрицы\n",
    "- Скелетное разложение\n",
    "- Малоранговая аппроксимация\n",
    "- Сингулярное разложение (SVD)\n",
    "- Приложения SVD"
   ]
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    "## Матрицы и линейные пространства\n",
    "\n",
    "Матрица может быть рассмотрена как последовательность векторов, которые являются её столбцами:\n",
    "\n",
    "$$\n",
    "   A = [a_1, \\ldots, a_m], \n",
    "$$\n",
    "\n",
    "где $a_m \\in \\mathbb{C}^{n\\times 1}$.  \n",
    "\n",
    "Умножение матрицы на вектор эквивалентно вычислению линейной комбинации столбцов матрицы\n",
    "\n",
    "$$\n",
    "   y =  Ax \\quad \\Longleftrightarrow \\quad y = a_1 x_1 + a_2 x_2 + \\ldots +a_m x_m.\n",
    "$$"
   ]
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   "source": [
    "## Линейная зависимость\n",
    "\n",
    "**Определение.** Векторы $a_i$ называются **линейно зависимыми**, если существуют коэффициенты $x_i$ не равные нулю одноврменно такие что\n",
    "\n",
    "$$\\sum_i a_i x_i = 0,$$\n",
    "\n",
    "или в матричной форме\n",
    "\n",
    "$$\n",
    "   Ax = 0, \\quad \\Vert x \\Vert \\ne 0.\n",
    "$$\n",
    "\n",
    "В этом случае говорят, что у матрицы $A$ нетривиальное **ядро** (kernel или nullspace), которое обознаается $N(A)$ (или $\\text{ker}(A)$).\n",
    "\n",
    "Векторы, которые не являются линейно зависимыми называются **линейно независимыми**."
   ]
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   "source": [
    "## Линейное пространство\n",
    "\n",
    "**Линейное пространство** натянутое на векторы $\\{a_1, \\ldots, a_m\\}$ определяется как множество всех возможных векторов вида\n",
    "\n",
    "$$\n",
    "   \\mathcal{L}(a_1, \\ldots, a_m) = \\left\\{y: y = \\sum_{i=1}^m a_i x_i, \\, \\forall x_i, \\, i=1,\\dots, n \\right\\}, \n",
    "$$\n",
    "\n",
    "В матричной форме линейное пространство – это множество всех $y$ таких что\n",
    "\n",
    "$$y = A x.$$\n",
    "\n",
    "Это множество также называется областью значений (range) или образом (image) матрицы и обозначается $\\text{range}(A)$ (или $\\text{im}(A)$) соответственно."
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   "source": [
    "## Размерность линейного пространства\n",
    "\n",
    "Размерность линейного пространства $\\text{im}(A)$ обозначается как $\\text{dim}\\, \\text{im} (A)$ и является минимальным числом векторов необходимых для представления каждого вектора из $\\text{im} (A)$.\n",
    "\n",
    "Размерность $\\text{im}(A)$ напрямую связана с понятием **ранга матрицы**.\n"
   ]
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   "source": [
    "## Ранг матрицы \n",
    "\n",
    "**Определение.** Рангом матрицы $A$ называется максимальное число линейно независимых *столбцов* матрицы $A$.\n",
    "\n",
    "Вы также можете использовать линейные комбинации *строк* для определения ранга, то есть формально существует два ранга: строчный и столбцовый.\n",
    "\n",
    "**Q**: как связаны ранг матрицы и $\\text{dim}\\, \\text{im} (A)$?"
   ]
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   "source": [
    "**Теорема**  \n",
    "Размерность пространства столбцов матрицы совпадает с размерностью пространства её строк.\n",
    "\n",
    "[Доказательство](https://ocw.mit.edu/courses/mathematics/18-701-algebra-i-fall-2010/study-materials/MIT18_701F10_rrk_crk.pdf)\n",
    "\n",
    "- В матричной форме этот факт может быть выражен следующим образом $\\mathrm{dim}\\ \\mathrm{im} (A) = \\mathrm{dim}\\ \\mathrm{im} (A^\\top)$.\n",
    "\n",
    "- Таким образом, ранг матрицы определяется однозначно!"
   ]
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   "source": [
    "## Матрица полного ранга\n",
    "\n",
    "\n",
    "Матрица $A \\in \\mathbb{R}^{m \\times n}$ называется матрицей полного ранга, если $\\mathrm{rank}(A) = \\min(m, n)$."
   ]
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   "source": [
    "Пусть имеется линейное пространство, натянутое на $n$ векторов.\n",
    "Пусть эти векторы сгенерированы из стандартного нормального распределения $\\mathcal{N}(0, 1)$.\n",
    "\n",
    "**Q**: Какова вероятность, что размерность этого подпространства $m < n$?\n",
    "\n",
    "**A**: Случайная матрица является полноранговой с вероятностью 1."
   ]
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   "source": [
    "## Снижение размерности\n",
    "\n",
    "- Многие данные в реальном мире лежат в пространстве высокой размерности. Например, изображения ($512\\times 512$ пикселей), тексты, графы, etc.\n",
    "- Однако, обрабатывать данные высокой размерности – сложная задача!\n",
    "\n",
    "**Q**: Возможно ли уменьшить размерность данных, сохранив при этом важные отношения между объектами, например расстояние?"
   ]
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   "source": [
    "**[Лемма Johnson'a–Lindenstrauss'a](http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf)** \n",
    "\n",
    "Пусть $N\\gg 1$. Дано $0 < \\epsilon < 1$, $m$ точек из $\\mathbb{R}^N$ и $n > \\frac{8 \\log m}{\\epsilon^2}$ (мы хотим, чтобы $n\\ll N$).\n",
    "\n",
    "Тогда существует **линейное** отображение $f: \\mathbb{R}^N \\rightarrow \\mathbb{R}^n$ такое что выполнено следующее неравенство:\n",
    "\n",
    "$$(1 - \\epsilon) \\Vert u - v \\Vert^2 \\leq \\Vert f(u) - f(v) \\Vert^2 \\leq (1 + \\epsilon) \\Vert u - v \\Vert^2.$$\n",
    "\n",
    "\n",
    "- Эта лемма утверждает, что существует линейное отображение из пространства большой размерности в пространство малой размерности такое что расстояния между точками в этих пространствах практически одинаковы.\n",
    "- Утверждение не практичное из-за наличия зависимости от $\\epsilon$.\n",
    "- Эта лемма не даёт способа построения $f$, но гарантирует, что такое отображение существует."
   ]
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   "source": [
    "## Скелетное разложение\n",
    "\n",
    "Очень полезным для вычисления ранга матрицы является **скелетное разложение** матрицы.\n",
    "Это разложение объясняет почему и как матрицы малого ранга могут быть сжаты.\n",
    "\n",
    "Графически скелетное разложение можно представить следующим образом:  \n",
    "\n",
    "<img src=\"./skeleton.png\" width='50%'>\n",
    "\n",
    "или в матричной форме\n",
    "\n",
    "$$\n",
    "   A = C \\widehat{A}^{-1} R,\n",
    "$$\n",
    "\n",
    "где $C$ образуют некоторые $r=\\mathrm{rank}(A)$ столбцов матрицы $A$, $R$ образуют некоторые $r$ строк матрицы $A$ и  $\\widehat{A}$ **невырожденная** подматрица, образованная пересечением строк и столбцов из $C$ и $R$."
   ]
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   "source": [
    "### Замечание\n",
    "\n",
    "Пока мы ещё не определили формально, что такое обратная матрица, поэтому напоминаем:\n",
    "\n",
    "- Матрица, обратная к матрице $P$ – это такая матрица $Q = P^{-1}$ что $ P Q = QP = I$.  \n",
    "- Если матрица квадратная и имеет полный ранг, то обратная матрица существует."
   ]
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   "source": [
    "### Доказательство существования скелетного разложения\n",
    "\n",
    "* Пусть $C\\in \\mathbb{C}^{n\\times r}$ – $r$ столбцов взятых из невырожденной подматрицы $\\widehat{A}$, поэтому они линейно независимы. \n",
    "* Возьмём произвольный столбец $a_i$ матрицы $A$. Тогда $a_i$ может быть представлен как линейная комбинация столбцов $C$, то есть $a_i = C x$.  \n",
    "\n",
    "* $a_i = C x$ – это $n$ уравнений. Возьмём $r$ из них, которые соответствуют строкам матрицы $\\widehat{A}$, и получим систему уравнений\n",
    "\n",
    "$$\\widehat{r} = \\widehat{A} x \\quad \\Longrightarrow \\quad x = \\widehat{A}^{-1} \\widehat r$$ \n",
    "\n",
    "Таким образом, $a_i = C\\widehat{A}^{-1} \\widehat r$ для каждого $i$ и \n",
    "\n",
    "$$A = [a_1,\\dots, a_m] = C\\widehat{A}^{-1} R.$$"
   ]
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   "source": [
    "### Подробнее про скелетное разложение \n",
    "\n",
    "Любая матрица ранга $r$ может быть записана в форме\n",
    "\n",
    "$$A = C \\widehat{A}^{-1} R,$$\n",
    "\n",
    "где $C$ – матрица $n \\times r$, $R$ – матрица $r \\times m$ и $\\widehat{A}$ размера $r \\times r$, или \n",
    "\n",
    "$$\n",
    "   A = U V,\n",
    "$$\n",
    "где $U$ и $V$ не единственны, например $U = C \\widehat{A}^{-1}$, $V=R$.\n",
    "\n",
    "Форма $A = U V$ является стандартной формой записи скелетного разложения.\n",
    "\n",
    "Таким образом, каждая матрица ранга $r$ может быть записана как произведение \"высокой\" матрицы $U$ и \"короткой\" матрицы $V$."
   ]
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   "source": [
    "В индексной форме его можно записать как \n",
    "\n",
    "$$\n",
    "   a_{ij} = \\sum_{\\alpha=1}^r u_{i \\alpha} v_{\\alpha j}.\n",
    "$$\n",
    "\n",
    "Для ранга 1, получим\n",
    "\n",
    "$$\n",
    "   a_{ij} = u_i v_j,\n",
    "$$\n",
    "\n",
    "то есть наблюдаем разделение индексов, а матрица ранга $r$ – это сумма матриц ранга 1."
   ]
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   "source": [
    "### Хранение и быстрое выполнение матрично-векторных операций\n",
    "\n",
    "Отметим, что для матрицы ранга $r$\n",
    "\n",
    "$$A = U V$$\n",
    "\n",
    "можно хранить только $U$ и $V$, то есть всего лишь $(n+m) r$ элементов, что может быть использовано для сжатия. Мы также можем вычислить произведение матрицы $A$ ранга $r$ на вектор $x$ гораздо быстрее:\n",
    "\n",
    "* Умножение $y = Vx$ стоит $\\mathcal{O}(mr)$ операций.\n",
    "* Умножение $z = Uy$ стоит $\\mathcal{O}(nr)$ операций.\n",
    "\n",
    "Это же применимо для сложения, поэлементного умножения и других операций.\n",
    "Например, для сложения:\n",
    "\n",
    "$$\n",
    "    A_1 + A_2 = U_1 V_1 + U_2 V_2 = [U_1|U_2] [V_1^\\top|V_2^\\top]^\\top\n",
    "$$"
   ]
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     "text": [
      "39.2 ms ± 6.77 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n",
      "67.5 µs ± 95.4 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "# A fast matrix-by-vector product demo\n",
    "import numpy as np\n",
    "n = 10000\n",
    "r = 10\n",
    "u = np.random.randn(n, r)\n",
    "v = np.random.randn(n, r)\n",
    "a = u.dot(v.T)\n",
    "x = np.random.randn(n)\n",
    "%timeit a@x\n",
    "%timeit u@(v.T@x)"
   ]
  },
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   "source": [
    "## Вычисление ранга матрицы\n",
    "Мы также можем попробовать вычислить ранг матрицы с помощью функции из NumPy: ```np.linalg.matrix_rank```"
   ]
  },
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rank of the matrix: 1\n",
      "Rank of the matrix: 50\n"
     ]
    }
   ],
   "source": [
    "#Computing matrix rank\n",
    "import numpy as np\n",
    "n = 50 \n",
    "a = np.ones((n, n))\n",
    "print('Rank of the matrix:', np.linalg.matrix_rank(a))\n",
    "b = a + 1e-5 * np.random.randn(n, n)\n",
    "print('Rank of the matrix:', np.linalg.matrix_rank(b, tol=1e-8))"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "### <font color='red'> Итак, малое возмущение матрицы может очень сильно повлиять на ранг! </font>"
   ]
  },
  {
   "cell_type": "markdown",
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   },
   "source": [
    "### Неустойчивость вычисления ранга матрицы\n",
    "\n",
    "Для любой матрицы $A$ ранга $r < \\min(m, n)$ существует матрица $B$ такая что её ранг равен $\\min(m, n)$ и\n",
    "\n",
    "$$\n",
    " \\Vert A - B \\Vert = \\epsilon.\n",
    "$$\n",
    "\n",
    "**Q**: Значит ли это, что численный ранг матрицы не имеет никакого смысла? (Малое возмущение приводит к полному рангу!)\n",
    "\n",
    "**A**: Нет. Нам нужно найти матрицу $B$ такую что $\\|A-B\\| = \\epsilon$ и ранг $B$ минимален. Таким образом, мы можем вычислить ранг матрицы только с заданной точностью $\\epsilon$.\n",
    "Один из подходов к вычислению ранга матрицы – это сингулярное разложение матрицы (SVD)."
   ]
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   "source": [
    "## Малоранговая аппроксимация\n",
    "\n",
    "Важная задача во многих прилоджениях – это найти малоранговое приближение для данной матрицы с точностью $\\epsilon$ или рангом $r$. <br>\n",
    "\n",
    "Примеры:\n",
    "* метод главных компонент (PCA)\n",
    "* факторный анализ\n",
    "* линейная задача наименьших квадратов\n",
    "* латентный семантический анализ\n",
    "\n",
    "Эти задачи могут быть решены с помощью SVD."
   ]
  },
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   "source": [
    "## Сингулярное разложение\n",
    "\n",
    "Для вычисления малорангового приближения нам нужно уметь вычислять **сингулярное разложение** (SVD).\n",
    "\n",
    "**Теорема** Любая матрица $A\\in \\mathbb{C}^{n\\times m}$ может быть представлена в виде произведения трёх матриц:  \n",
    "\n",
    "$$\n",
    "   A = U \\Sigma V^*,\n",
    "$$\n",
    "\n",
    "где $U$ – унитарная матрица $n \\times K$, $V$ – унитарная матрица $m \\times K$, $K = \\min(m, n)$, <br> $\\Sigma$ – диагональная матрица с неотрицательными элементами на диагонали $\\sigma_1 \\geq  \\ldots, \\geq \\sigma_K$. <br>\n",
    "Более того, если $\\text{rank}(A) = r$, тогда $\\sigma_{r+1} = \\dots = \\sigma_K = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Доказательство\n",
    "* Матрица $A^*A$ эрмитова, поэтому диагонализуема в унитарном базисе (подробнее будет рассказано на следующих лекциях).\n",
    "* $A^*A\\succeq 0$ (неотрицательно определена), поэтому собственные значения неотрицательны.\n",
    "Поэтому существует унитарная матрица $V = [v_1, \\dots, v_n]$ такая что\n",
    "\n",
    "$$\n",
    "    V^* A^* A V = \\text{diag}(\\sigma_1^2,\\dots, \\sigma_n^2), \\quad \\sigma_1\\geq \\sigma_2\\geq \\dots \\geq \\sigma_n \\geq 0.\n",
    "$$\n",
    "\n",
    "Пусть $\\sigma_i = 0$ для $i>r$, где $r$ некоторое целое число. <br>\n",
    "Пусть $V_r= [v_1, \\dots, v_r]$, $\\Sigma_r = \\text{diag}(\\sigma_1, \\dots,\\sigma_r)$. Таким образом,\n",
    "\n",
    "$$\n",
    "    V^*_r A^* A V_r = \\Sigma_r^2 \\quad \\Longrightarrow \\quad (\\Sigma_r^{-1} V_r^* A^*) (A V_r\\Sigma_r^{-1} ) = I.\n",
    "$$\n",
    "\n",
    "В результате для матрицы $U_r = A V_r\\Sigma_r^{-1}$ выполнено $U_r^* U_r = I$ и следовательно её столбцы ортогональны. <br>\n",
    "Присоединим к матрице $U_r$ произвольные ортогоналоьные столбцы, которые также ортогональны столбцам $U_r$ и обозначим такую матрицу $U$.\n",
    "Тогда\n",
    "\n",
    "$$\n",
    "    AV = U \\begin{bmatrix} \\Sigma_r & 0 \\\\ 0 & 0 \\end{bmatrix}\\quad \\Longrightarrow \\quad U^* A V = \\begin{bmatrix}\\Sigma_r & 0 \\\\ 0 & 0 \\end{bmatrix}.\n",
    "$$\n",
    "\n",
    "Поскольку умножение на невырожденную матрицу не меняет ранга $A$, получим $r = \\text{rank}(A)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Следствие 1**: $A = \\displaystyle{\\sum_{\\alpha=1}^r} \\sigma_\\alpha u_\\alpha v_\\alpha^*$ или поэлементно $a_{ij} = \\displaystyle{\\sum_{\\alpha=1}^r} \\sigma_\\alpha u_{i\\alpha} \\overline{v}_{j\\alpha}$\n",
    "\n",
    "**Следствие 2**: $$\\text{ker}(A) = \\mathcal{L}\\{v_{r+1},\\dots,v_n\\}$$\n",
    "$$\\text{im}(A) = \\mathcal{L}\\{u_{1},\\dots,u_r\\}$$\n",
    "$$\\text{ker}(A^*) = \\mathcal{L}\\{u_{r+1},\\dots,u_n\\}$$\n",
    "$$\\text{im}(A^*) = \\mathcal{L}\\{v_{1},\\dots,v_r\\}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Теорема Eckart'a-Young'a\n",
    "\n",
    "Наилучшая малоранговая аппроксимация может быть вычислена с помощью SVD.\n",
    "\n",
    "**Теорема:** Пусть $r < \\text{rank}(A)$, $A_r = U_r \\Sigma_r V_r^*$. Тогда\n",
    "\n",
    "$$\n",
    "    \\min_{\\text{rank}(B)=r} \\|A - B\\|_2 = \\|A - A_r\\|_2 = \\sigma_{r+1}.\n",
    "$$\n",
    "\n",
    "Похожий результат справедлив и для $\\|\\cdot\\|_F$, но $\\|A - A_r\\|_F = \\sqrt{\\sigma_{r+1}^2 + \\dots + \\sigma_{\\min (n,m)}^2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Доказательство  \n",
    "\n",
    "- Поскольку $\\text{rank} (B) = r$, то справедливо равенство $\\text{dim}~\\text{ker}~B = n-r$. \n",
    "- Тогда существует $z\\not=0$ такой что $z\\in \\text{ker}(B) \\cap \\mathcal{L}(v_1,\\dots,v_{r+1})$ (так как $\\text{dim}\\{v_1,\\dots,v_{r+1}\\} = r+1$).\n",
    "- Зафиксируем $\\|z\\| = 1$. Тогда,\n",
    "\n",
    "$$\n",
    "    \\|A-B\\|_2^2 \\geq \\|(A-B)z\\|_2^2 = \\|Az\\|_2^2 = \\| U\\Sigma V^* z\\|_2= \\|\\Sigma V^* z\\|_2 = \\sum_{i=1}^{n} \\sigma_i^2 (v_i^*z)^2 =\\sum_{i=1}^{r+1} \\sigma_i^2 (v_i^*z)^2 \\geq \\sigma_{r+1}^2\\sum_{i=1}^{r+1} (v_i^*z)^2 = \\sigma_{r+1}^2\n",
    "$$\n",
    "\n",
    "так как $\\sigma_1\\geq \\dots \\geq \\sigma_{r+1}$ и $$\\sum_{i=1}^{r+1} (v_i^*z)^2 = \\|Vz\\|_2^2 = \\|z\\|_2^2 = 1.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Основной результат о малоранговой аппроксимации\n",
    "\n",
    "**Следствие:** вычисление наилучшей аппроксимации ранга $r$ эквивалентно использованию SVD с $\\sigma_{r+1}= 0, \\ldots, \\sigma_K = 0$. Тогда ошибка аппроксимации равна \n",
    "\n",
    "$$\n",
    "   \\min_{A_r} \\Vert A - A_r \\Vert_2 = \\sigma_{r+1}, \\quad \\min_{A_r} \\Vert A - A_r \\Vert_F = \\sqrt{\\sigma_{r+1}^2 + \\dots + \\sigma_{K}^2}\n",
    "$$\n",
    "\n",
    "поэтому так важно смотреть на степень убывания сингулярных чисел матрицы."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вычисление SVD\n",
    "\n",
    "Алгоритмы для вычисления SVD довольно нетривиальны и будут рассмотрены на следующих лекциях.\n",
    "\n",
    "Но для вычислений мы уже можем использовать NumPy!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rank of the matrix: 1\n",
      "Rank of the matrix: 50\n"
     ]
    }
   ],
   "source": [
    "#Computing matrix rank\n",
    "import numpy as np\n",
    "n = 50 \n",
    "a = np.ones((n, n))\n",
    "print('Rank of the matrix:', np.linalg.matrix_rank(a))\n",
    "b = a + 1e-5 * np.random.randn(n, n)\n",
    "print('Rank of the matrix:', np.linalg.matrix_rank(b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.7459639108605272e-06\n",
      "2.7459639108671015e-06\n"
     ]
    }
   ],
   "source": [
    "u, s, v = np.linalg.svd(b)\n",
    "print(s[1]/s[0])\n",
    "r = 1\n",
    "u1 = u[:, :r]\n",
    "s1 = s[:r]\n",
    "v1 = v[:r, :]\n",
    "a1 = u1.dot(np.diag(s1).dot(v1))\n",
    "print(np.linalg.norm(b - a1, 2)/s[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Разделение переменных для двумерной функции\n",
    "\n",
    "Мы можем использовать SVD для вычисления аппроксимации матриц, полученных из вычисления некоторой функции в точках сетки\n",
    "\n",
    "$$a_{ij} = f(x_i, y_j),$$\n",
    "\n",
    "где $f$ заданная функция, а $x_i, \\; i = 1, \\ldots, n$ и $y_j, \\; j = 1, \\ldots, m$ некоторые одномерные сетки."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Singular value index, $i$')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rc(\"text\", usetex=True)\n",
    "import seaborn as sns\n",
    "sns.set_style(\"whitegrid\")\n",
    "sns.set_context(\"talk\")\n",
    "# plt.xkcd()\n",
    "n = 1000\n",
    "a = [[1.0/(i+j+1) for i in range(n)] for j in range(n)] #Hilbert matrix \n",
    "a = np.array(a)\n",
    "u, s, v = np.linalg.svd(a)\n",
    "plt.semilogy(s[:30]/s[0])\n",
    "plt.ylabel(r\"$\\sigma_i / \\sigma_0$\", fontsize=24)\n",
    "plt.xlabel(r\"Singular value index, $i$\", fontsize=24)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Теперь попробуем аппроксимировать заданную функцию"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Approximation error = 3.6130603628595414e-10\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Singular value index, $i$')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "n = 128\n",
    "t = np.linspace(0, 5, n)\n",
    "x, y = np.meshgrid(t, t)\n",
    "f = 1.0 / (x + y + 0.5) # test your own function. Check 1.0 / (x - y + 0.5)\n",
    "u, s, v = np.linalg.svd(f, full_matrices=False)\n",
    "r = 10\n",
    "u = u[:, :r]\n",
    "s = s[:r]\n",
    "v = v[:r, :] # Mind the transpose here!\n",
    "fappr = u.dot(np.diag(s).dot(v))\n",
    "er = np.linalg.norm(fappr - f, 'fro') / np.linalg.norm(f, 'fro')\n",
    "print(\"Approximation error =\", er)\n",
    "plt.semilogy(s/s[0])\n",
    "plt.ylabel(r\"$\\sigma_i / \\sigma_0$\", fontsize=24)\n",
    "plt.xlabel(r\"Singular value index, $i$\", fontsize=24)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "И несколько 3D графиков"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+/Pm8q4tkI5fdqIT78vTp0zT5bGhoqKhElUL2UxRFCIIg+90jjzyC6elpet5IjK3b7c5qg5WeVzUoxbOBXJfpJeXyYWJiAjzP06oYpAQdeQ4ePnwYp0+fpr/VgQMHUhzOqakpxedOzWsoX2w2G53BHR0dxeOPP46urq6UWV25YxkYGMCFCxcwPj6O4eFhWK3WlHUKOSae55/keV7E3fawN3meF0n5wG379xSS3bpeBvBPLpcr08t+ClwOZ0WZJ8OoaKampnD06NGUzEYyVQgAly9fhiiKiMViCIVC4DgOer0eOp0OWq0WGg17z2FUHdxu70Ch8Dz/1wAcaUY8xRaTmEI1p/cZlYEgCIhEIgiHwzAYDNQOa7VacFzVXtY1w8GDB7PGtjJSKOkFyzyTPQCJWZHGI5FpAbfbDUEQEI1GEY/HIQgC4vE4QqEQAoEA/H4/tra2EIlEEI/HFauzDAaDwVCOKIpIJBKIRqNIJBLUkd3a2oLf74ff70cwGEQ0Gs2ozjJKy/T0NB555JHd3g3GNsyB3QMcPnwY09PTtEMOcHf64Atf+AI1mBzHQaPRQKfTQafTQaPRQBRFakSJQxsKhZgRZTAYDJUQRRHxeJzaVY7joNVq6UyYRqNBIpFAOBxGIBCAz+dDIBBAOBxmwkIZsdlsZQsDYOSGJXHtAZxOJ65du0a7Uvl8Ptx///34jd/4DXz+85+HIAjQaDTweDxYX19HXV0dLBYL6uvr6dSVKIoQRREbGxtYW1uD0+mkxlWv17MpLgaDwSgAEr4Vj8epiJBIJOj3HMfRz8nygiBgdnYWGo0G+/btg1arpcIDCftitlh9WGOayoLFwO5BiKoaj8eh1WoBAEtLS/B4PDAYDDSUAACMRiPMZjPMZjPq6urg8/mwvLyM3t5eaLVauhxRDIgR1el01PAyGLtArV14zBbXIERVFUWR2kzyeTQapfZZjrm5OWg0GnR0dKQk32o0GqreEpvM7DBjlyjphccU2D0GedufmZmBRqNBV1cX5ufnsbW1hcbGRjQ2JusJk7CBra0trK+vY319nRpFIJloYDAYUtTZRCKBeDwOACmhCEQVYEaUwWAwkiQSCfj9fty8eRNOpzPf2pkUjUazQ52NxWKIxWIpYWHEoWXqLKNWYA7sHoIYNjI9JQgCZmZmEI/H0dnZCbvdjlgsBgAwmUwwmUxobm5GPB5HMBikDi2QfPuvq6uj6qzRaIROl7yciBGNRqOIRqM71FlmRBkMxl6FvOzHYjHV8wiIrZUKCyQZLBKJMHWWUVMwB3aPIDWYJMYqGo1Cp9Ohr68P9fX1GRMBdDodbDYbbDYbNjc3sby8DIfDgXA4DI/HA4/HA41GA7PZjPr6epjNZuqoZlJnmRFlMBh7DTIDRkSEYkoUchyXNXkrU+xsPB7foc5KE3eZLWZUC8yBrXGkb/uiKILjOKyuriIajUKj0WD//v00LEAJxLg5HA4YDAYkEokUdZb0uybqbH19PUwm0w51lk1xMRiMvQSxe0R1TZ/6LzVEnSXjZVJn02fKGIxKhTmwNUx6disAzM/Pw+fzQavVwmAw5OW8yqHVamG1WmG1WmlyGHFopepsfX09VWeJo8qmuBgMxl6AhFSRGbDdtmn5qrOsygyjEmEObI0ijXflOA7xeBxzc3MIh8Noa2uD3+8v6q1fbl2O42A0GmE0GtHY2EjVWeLQBgIBAIDBYMgaO+vxeOD1euF0OlOmt9gUF4PBqCak4VNyziv5dyZ7Wi7k1FlRFHHr1i0YDAa0trZCo9GwDo2MioI5sDVIerxrMBjE3NwcRFFET08PrFYrtra2UmoNKiUfo5quzkajUWxtbSEYDMLr9cLr9YLjOKrMEnWW4zgkEomURgpsiovBYFQTpDmBNPZfTaeU47iSNJORlj8k5bmITSazeentxpk6y9gNmANbQ6THu2o0Gni9XiwuLkKv16O7uxtGozFl+XLBcRzq6upQV1eHxsZGCIKQEjtLqhsQo0j2T5oMxqa4GAxGNZCerJWpJnY2BbZSILaWCArkPyIsyNUAZ8ICoxwwB7ZGkDOYi4uL8Hg8MJvN6O7uTimKvduOnkajgcVigcVioftOHNlgMAgAuH37Nkwm0w51Vs6IEgPLprgYDMZuIg3fKkZ1rUSnVuqIy1WZId+zGuCMcsAc2Bog3XlNJBK0OUFTUxP27dtX0QaE4zgYDAYYDAY0NDTQxgk2mw3BYBCrq6tYXV2FXq+n4QZybW7ljChrc8tgMMpFevhWLpuTSYGtFlslTQYjx5CtBjjr0MhQE+bAVjnp2a2RSARzc3OIxWLo7OxEQ0OD7Hq5aghmohxTXmSMlpYWaDQaRKNRGm7g8/mwubkJjuOoOltfXw+DwQCdTqfYiDJ1lsFgqIVc+FapnbRCbXipkCqzSmqAM3WWUSzMga1SpPVUifPq9/sxPz8PjUZDmxPkS6UZRQBUnXU4HBAEAaFQiDq0q6urAJLNFkiogclkymlE2RQXg8FQA7nmBErtSSZBYGNjA4FAIGW2qdrIVKqL1QBnqAVzYKsQ4pBJS5ysra1heXkZRqMRPT09Oeu7VqKjqgTS8ctsNqOlpYXGzgaDQfj9fmxubgJASuwsUWeBu0Y0FArRjmI2m421uWUwGHkjCAJ8Ph8WFhbQ3t5ekGggRRRFmrsAAD6fD8BOe1aN9ilTm9u1tTUEAgF0dnayGuCMvGAObJUhbU4QDocBAHfu3MHm5ibsdjs6OzvLMj1eKc6vXq+Hw+GAw+GAKIoIhULUoV1bW8Pa2hq0Wi01/kTNIM0TWJtbBoNRCCRkIBaLIRQKFVXSipTccrvdNHfB4XDQ0oNbW1vUnul0OtTX1yMej1eMHc4XqTorCAJ9lrE2t4x8YA5sFZHenAAAAoEABEFAW1sbmpub85q6KiYGthIhNWWJChKPx6nxDwQCVM0gzRYIrM0tg8FQily8a6EQWxKPxzEzM0NzF+x2O6LRKEwmE0wmE5qbm3fYM+Iwu93uqlZnSZ3ZTG1u5fIYWJUZBsAc2KohPbs1FAohGo0CALq7u2Gz2XZ5DysPnU4Hu90Ou90OURQRDofpA2BjYwMAsLa2hlAoRB8AclNcrM0tg8EA5ONdpTGehUJabvf29sJsNsuquen2bH5+HpFIBIlEIkWdlc42VaOTl2+bW6bO7l2YA1vhyL3tb2xsYGFhgWbiF+K8VmsMbKGQc0XUjFAoBLfbDYPBgK2tLfj9fgBJdZaU6pJrc8uMKIOxN1HanCCf7ZFYV61Wi/7+fhgMBkXrShXJ3t7elFwAuUot0jra1UamNresBjiDObAVjJzBXFpawvr6OsxmMy2UXYpxq9HQ5QMxiA6HA1arFeFwmFY28Hg8VBGRqhlSZzaTOsumuBiM2iNbc4JCSguKooiFhQV4vV4ASTuU7rzms71MuQCkUsvq6mrFqrP5PG+UNlJgbW73BsyBrVBEUaT1XYGkAZUG+O/btw8zMzMFb7+S68CWA7L/xOARdbapqQmJRIKqGVJ1tq6ujj4A8lFnmRFlMKqXXM0J8rWJ0mSt5uZmrK2tFWQbMtlwaS6AtFJLpjra1a7OShspZFJnWQ3w2oQ5sBVItuYEHR0daGxsBLD3wgDKhVarhc1mg81mo8aQOLRSdVbaFYyULUs3oj6fD6FQCA0NDdTpZeosg1H5lKI5QTgcxuzsLOLxOLq6uuBwOLC2tpbREVUDqTpLSggSeybtciito12N9imXOuv3+6HRaNDQ0MBqgNcIzIGtIOSaEwQCAbjdbnAcRwP8CcU4sMz5VQbHcbRqAVFniTIbDAYRCAQAJJstpD8AtFottra24PV6YbVaEQqFWJtbBqMKkJYrJApfpntUqQLr9/vhdrt3NJop570vDYsCkKLObm5uYmNjI0WdLcczolQha+nqrN/vpyUVWZvb2oA5sBUCqQNIavtxHJfSnKC7u1txgD8jN9IQgnzQarWwWq2wWq00zIM8ALxeL7xeL53CI3HKAOjbPsDa3DIYlUx6ucJc92MuB1YURayvr2NpaanibHkmdVba5TAQCGBlZaXq1Vnyf71ez9rc1gjMga0A0t/2gbvNCWw2G7q6umSNBsdxRRfPzpdaiYFVA47jUFdXh7q6OjQ2NkIQBKrOkv8I6+vrsFgsrM0tg1HB5Ip3zYacTRQEAYuLi/B6vRlteTGzYWra4XR1NhqNYm5uDhzHpaiz5OWcxM5WC1Kll7W5rQ2YA7vLpGe3xuNxzM3NIRQKobW1FS0tLVmnrooJIWCoi0ajgcVigcVioS8lq6urO5InSIIFKTyebkQzqbPMiDIYpaGYeNdMyxFbHgwG0dLSgtbWVlXv3VLbAYPBAK1WC6PRiLa2NgSDwZQXdLIMsWWFqrPlqnqTbZxMbW5ZDfDKhjmwu0g0GkU4HKY3TigUwtzcHARBYM0JSkyhIQRK4TgOBoMBJpMJW1tb6OvrSynVRaboSPIEcWpzqbPMiDIY6iKKIoLBIBKJBPR6fd4viXKzUnLJWtnWr/QZLbmXc7nY2WpVZ6WwRgrVA3NgdwHimHg8Hty5cwf33HMP/H4/FhYWoNPp0N/fn9LqNBMsiavyIedYq9XSBwAAGjsbDAZlkyfIA4C1uWUwSgeJY5+bmwMA9PX1FXwPkXvd5/Nhfn5+R7JWrUBezg0GAxoaGjKGTqUntu62bSpU6WVtbisX5sCWmfTmBECynenGxgbMZjOcTid1WnKhhhOa703NYmDzI9N5Sn8AyJW2SS88nm2Ki2A0Gqnju9sPDAajkpGWKyS2tFAHB0je66urqzTxtqenR5EKWSkxsIWSrs5Go1Hq0EoTW4kdS1dnyxlCUCxK1FlBEKDVamEymZg6W2KYA1tGpCWygLuGb2NjA42NjWhvb8/7QmcxsIVR6hCCdLKNk620DYmdBZCizhoMhhR1NhAIYHFxEe3t7TQWjU1xMRg7kStXWExCLLmvNjc3EQ6HsybeFrKvucp3VRLSxFY5dVau7GA5nXC1z5mcOru8vIxoNEqvAdbmtnQwB7ZMpDcniEajWFpaAgA0Nzdj3759eW+zEg0YI5VC1IVsbSHX1tawtrYGnU6X0kiBjEGcVdbmlsHYSXq5QuK8FqOCkhj1cDicM/FWjkoO5yp2v+TU2fSygwBopQOz2ax4BjJfSq30SmvIEgGBtbktLcyBLTHSCzi9OQGh0GQtNaoQlGv6hlEY0qoFpC0kUTQCgQB8Ph8A0LqSsViMtbllMGTI1pygUFtKkrWApB1vbW1VdZ8rAbVsQ3rZQdIUZnV1FfF4HMvLywB2tuyuNttEnqnpjRRYm1v1YQ5sCZG+7QPJG1ha0Lq5uRnz8/MV+/YtR63EwJYrhEDtFwS9Xg+73Q673Q5RFBEOh2moAQAsLS1hdXU1Jd4sfYpLzoiyKS5GLZNerjD9nizEgZUmawFJx6tQqjkGtlBIUxi/349IJIKOjg6qzsq17C5Wnd3Ncl1SdVauygz5ntUAzw/mwJaI9GQtURSxsLCQ0pwgFArR7wpBLQWWUTpKaTRJ1QKTyQSDwYClpSU0NzcjEomkqLNGo5E6tEajkYYUZDKibIqLUUsobU6g1BaKoki7JJpMJnR3d+Pdd98tez5CLd2XmdTZ9NjZYtTZcjqwSrq3SdVZIHeHRqkTzEjCHNgSoLQ5QbFOZCXHThVDufpv1xLkeKxWKxobG1PU2WAwiPX1dayvr6ckjNXX19NpK6kRJeqsnBFl6iyjWsinOYFSWyoIAu2SaLfb0dnZWfQ9kWnsveKsyDmW6S27I5GIrDqbbsuyjVEu8h1LqswqqQHO1Nm7MAdWZdLf9rM1J9hNFbQaFNhy3KDVFkKQbRwgtec3UWeB5HVJHgDBYBB+vx/ATkWDtbll1ALpM2C5qnAouY5jsVjGLom1KiZUAhzHwWg0wmg0oqmpaYc6m8mWyf2muxVCkA+ZSnWl1wBPD/vai7aYObAqIfe2v7GxkbU5QTUqsNXg+FYS5TpPuWJ6tVotbDYbbDZbiqIRDAazxpvJtbldWVkBkKyewdrcMiqN9HKFShTSXLY0FAphdnYWiUQCTqcTdrs9r/VzIbcuS7CVR4k6q9VqU2yZ9EWjHKg5DpkJS68BvrW1hfX1ddgwcoz8AAAgAElEQVTtdlgslj3ZoZE5sCqQ/rbPcRyWl5extraWtTnBbjqwzBEtH7uhwGYjl6IhjTeT9jknRjQcDmed4tprRpRROaSXK1R6DWazpZubm5ifn6dCBJnVUAul+2g2mxGLxRCNRul6tWC/i3HU5WxZppkmAFRgKqVtKnXeA1FgiRNrNpsrps3t9evX8frrryMYDEKr1eKFF174ZQDvAnjT5XIF1R6PObBFkv62L4oi3G43AoFAzuYEajmRat4we8XpqNYqBNnGAQo7nnRFQ65WI1Fn6+vraaeZbG1uOY7D6uoqbDYb2traVD1WBiMduXKFxdZiFcVkZ62VlRWarJWps1a5nMm9Yp8LJX2mieQBkPa2m5ubCAQCOzocqkm5bT6xxVJ1VtrmNhaLwePxoKenR1GL+kJZXl7GW2+9hTt37iAcDmNjYwMAngJwE8BFAH+v9pjMgS2CWCyGUChE33ii0ShmZ2cRjUbR0dGBxsZGRdspd/aqGmMzlFHuEIJikcsGljZSIOoscVDr6+tT1FlpmS6fz4d4PM4cWEZJIY5KLBaDXq8vSHFKd0DzTdYqdjZM6brMgVWONA/A4XBgZmYGVqsVAGSrtJjNZtTV1RV9jncz70EudpY47X6/v6QOrNfrxWOPPZbi9/A8f8jlcgV4nv80z/MfcLlcb6g5JnNgC4C87W9sbGBubg59fX0Ih8Nwu93gOA59fX20LWg2ir3IpQpuIQa7mP1hjq9yypk4oPZYWq12Ryed+fl5AMkWyKTPubQrmMFgoEaUVS1glBISvuV2uxGNRnHPPfcUdA9Ibam0akxbWxuam5vLMlOTDpnpkOxlSfdhtyhnrC9xZtPVWVKlRavVFq3Olut4csV4k9hZgtpKczoGgwHf/e534XQ6YbVaSZnQh3ie3wAgAlhXe0zmwOaJNN6VXKTkQW40GtHd3U27IuVCjRjYYtZPX1cURWxubkIQBFgslqxxu9VOLYYQlHocos4SZaOtrY3GzpL/A8D6+jouXbqE9vZ2PPTQQ7j33nuZI8tQnfRyhWrcA8FgEG63W7ZqTDbUyEcgSNVfg8EAs9kMi8UCs9mi2ph7jWxVWpqbmxGPx1NmmYg6azKZqENrMBgUXV+VFjZWrmddY2MjQqEQvvvd76Kurg7hcBgAVgAMAmgC8GO1x2QObB6kJ2sRvF4vbDYbOjs783rLqYQyWoREIoH5+Xka9L6yskKnViwWi+Kbl7E7lFPFIFU2pH3OAdDY2Rs3buCHP/whAODSpUv43d/9XXz2s5/FM888k7KP4+PjsNlsmJ6ehtVqxalTp7KOOz4+DqfTCQBwu904duxYiY6QUenINScoxo6SF6zbt29nrBqTDbWcSaL+BoNBWK1WxONxGodOmpT4fD5FM3yMu+T6bXQ6nWyHw62tLaytrWFtbU2xOltpDixRakutwDocDjz11FN45513sLGxgYMHD+IP//AP3wCgatiAFObAKkSuOcHS0hIAwG63o6urq+Bp/N1WYGOxGGZnZxEOh9Ha2gqDwYBQKIRAIECnVnQ6Ha2vx1BOLSmwucYyGAwwGAwYHh7G/v378f3vfx83btzAO++8g0uXLuGrX/0qzQYeGxvDU089RRWu0dFRHD9+HM8//7zsmGNjY+ju7sbhw4cBAJOTkxgbG8vp9DJqi0zNCYpxIEVRpC/uRqMRPT09RbUszRey75FIBLOzs4jFYujs7ITJZALHcRAEAcFgkGaek+cO2cdIJFLVAgP5HcuB0iothaqz5VI7gfwV2HKd4/vuu4/+m+f5dgAel8sVKcVYzIHNgTTDOr05AVFibTZb0XFXxe5joWNHIhEsLi5CEASapZhIJFBfX4+mpqaUm9fn82FzcxNAUnUWRRFmszljZm4lU84QgnIYjkpwYKX09/fjs5/9LI4ePUqvI2mveL/fnzI9OzIyguHhYfh8Ptlp20uXLuHll1+mfw8ODuLZZ59lDuweIltzgkIdWEEQMD8/T8NfMpU8zEWxCmwikcDMzAwAoLe3FyaTCbFYDMDdKiEAUF9fD6fTia2tLWqLZ2dnodPptsMMkiXvWMhOKsXY+3R1VprUStRZIvCQEmuVEANLKKdTTcaTxG8/CuDvANwpxVjMgc2C3Nv+5uYm7ty5A51OB6fTibm5uV1TUNW4IOfn56HVatHX15diNAnSm1cQBAQCASwtLSEej9OC9gaDgRrPfPtTM9ShnNUOlDiwUqNJHr4Et9uNiYkJjIyMYGBgAACo07q5ubnDgXW73TscW5vNBp/Ph+npaboNRu2SqzlBIQ6kdOaJXE/F1CMt9B6Mx+NURe3p6UFdXR09ToJUJJB22PN4PGhtbaUO7cbGBk2qJDa5nGpyrUPObX19PVpaWhCLxVJqzpKXCvL/fGJn86XSQggIacljWgCxLIsXBbuyM0De9uPxOJ2iIs0J6uvr0d3dnVL7tVDUCP4vZH1SDkmv16Ovr0+RikpqgQJAU1MTzGYzAoGAbH9qi8VSkhp7alFOh6+WQgiUTkdlUwecTicuX76c4nhOTU3BZrPRGFcpbrc7YzINeVAwahe5eNd08rWjwWCQtvju6elBPB6Hz+craz6CKIpYWVlBJBKBRqNBf39/RmeTfi45dnIe7HY7HA4HDTUg6iBRlZW0WN1NymG7SqVC6vV6OBwOWtkgEAjQGc10dZbEzqqljldqCEEazIEtN9J4V2IYSYJTQ0MD2tvbodFoqFq5Ww4sIZ/1RVHE4uIivF4vAKCzs7OgEACO42AwGNDY2EjrhQaDQerQkpgyEidEWt1VmvHcjfI4pRqnmuKu0lXT8fFxPPfcc7LL+nw+GncmhahmjNokU7yrHMSOKrkPNjY26CwaSdbaLrpellquQGqlAdKKWYlSqslybNKkSlLyjsRtSlusEmfKbDbvmVCDckyjcxxHc0Sam5tRX1+/I/yOxNeS81/MM5E5sMyB3UH62340GsXc3BwikQja29vR1NREl93NKgLS8ZUirTRgtVrh9/uLvnkI6d2cSBZnIBCgb6J6vT7lTbTSnNlSUa7jrCQHNp8+9KOjozh58iRN0Eonk/qaKV6WUf2QeqzSVsXZrjlp8fZMyxHFkzTg6O7upk6jGuFcSteVVhpobW1FMBikxyndXqZxlO5PekMSuUQkUr+ZTHUzikNqH6XqrCAICIVCVCFfXV3F6uoq9Hp9Sg3tfJxMYmMr3IHlwBzY0iP3tr+1tYW5uTlwHIfe3l5aLohALpz0eKV8KFcIgTTeq729HQaDgaqkhYyZaxlpFieJE0qP05IqAeWO02J1YAsj38SBXMtNTExgcHAwo/MKJKdIC/mOUZ2kJ2uREK5s5LKFiUQCd+7cgc/nS5lFU7q+WkgrDXR1dcHhcCjKo5Amq6V/puTeT2+xKk1EkjpT0kSwSkoKVWMMoHwzbunjkNA6s9mcEjubTZ3N9UJBzptSMWGXBCMdgETOpYrY+J5HzmCur69jaWkJdXV16Onpkb2Y1DB65XBgQ6EQZmdnabyX1WqlMbDlUI/T30SlcVpkP+rq6qjxZMW586ec1Q4Add76p6amYLfbqfPqdrtht9t3qKoDAwM0XIB8R/7NErhqi/RyhfkojoC8PSOzaOFwGPv27UNTU9OO7aqhwOYSMoggAgB9fX00n6DcpCcikVCDra0t2pSH5DskEomiBJpKodwObC7k1Fm5FwrizMpVllDq+Ct1dEvEmy6XK557scLY8w4siRWSJmQtLi5iY2MDVqsVXV1dGRORdtuBVYLP56OVBvItzl0K0uO0IpHIjnZ+5LwGAoGqj9OqNQVWrczX6elpuN1uPPLII1SBePHFF2lZLLfbjampKYyMjAAAjhw5gitXrtC/r1y5giNHjqhyTIzKQEmyViYy2eL0ZK30ihjplCqEwOv1YmFhAXq9nlYaUGNcNSD1mxsaGqjAQHIZSGLS3Nwca2qjgEIcZak6CyDlhSK9soS0XXc+DuxuPUNdLtckAPA8zwHQuVwuVcMJ9rQDKwgCdV5JcwLSA7ulpQWtra1ZLxDyVlOpCuz6+joWFxdpcW5psla2dctlTEnQu9FoTKk56/V6EY1Gsbi4SKdViDqrVs3ZWkyuqiQHNttyPp8Pjz/+OHw+H0ZHR+nnNpuNOrBTU1MYHx+nDuupU6cwPj6OyclJbG5uYm5ujtWArRHySdbKhJw9kzqNvb29WV/eSxVCII27NZvN6O7u3vFSV0nOYLrAcPPmTej1eoiiuKOpjVpZ9bUUQkAoZpz0FwqpOksqS0ifg8R/kUNpYmMpmJ6exvDw8BcB7EcymYvjeX4NwDyAVwBsFuvQ7kkHlhTavXPnDsLhMLq7uxEOhzE7O4tEIgGn06k4tq4SHViiIns8HlitVjidTsVGpthYs2IgNWfj8TjW19fR2dlJb1ppzVnptEqxN2YlPTyKodwxsEodWDkF1maz4bXXXsu6/sjICHVeCax1bO1BZmFmZmbQ0NCAhoaGgq5jqV0SRZGWPDSbzYqaE6hhR9JtorTSgMPhQEdHR1ZHo1BK9TJOBJq6ujrs27dPtqmNVBms5KY25RQsAPWeK1J1loQ6kt8gGAwCAG7evJnzNyinAiuKIt566y185zvfAYCvAjACMACIAtgHoA5AEMA5AL9TzFh7zoGVZrdGo1GEQqGU5gT9/f20SLQSdtOBlSORSMDtdiMQCKCpqQn79u2TvZl2u4JCNsg+kZsSSJ1WIb3BpTe32WyuyJqztarAFlMHlsEAUuNdQ6EQrFZrwdcwWS+RSGBubg5+vx+NjY1ob29XnHgKqBdCIK000NbWhubm5qIqC2Sz4cQ5L6UjS8aRNrWRUwYrvalNNSu9HMelqLPz8/OIRqMwm807fgPyTCTXZTl/h2AwiFdffRVf/vKX8eyzz35Mbhme5+sAPMHz/KjL5Tpd6Fh7yoElbzDS5gSktFR6WRWlVJICK6000NHRgcbGxoL3qxKQ3nTSG5fUnCU3bXrN2VJ2P8mXWnVgiwkhYDCk8a7kxbNYOwqAPtTb29vR2NioShKYUsi6kUgEt2/fRjweVzybV+i4JPk1Ho8jFAoVtI1CkFMGC21qU0shBOW2ezqdDm1tbTSXhzwXvV4v3njjDXzta19DY2MjPvKRj+DRRx/Fxz72MSoKEaampvDiiy/i+eefzzne+Pg4bTbjdrtlZ8U4jsODDz6I7u5u4qgCgCj5Dy6XK8Lz/AUAjxV1/MWsXE2kZ7cKgkCD1OXKqiilUhxYuUoDSteV+7ySyVRzNr03dbbe4LXmYFWqA1uJqjhj98gU71qsHY1EIgCSymevTMnDXKihwAKgCU8cxymuNFDMfdvc3Fz0NoqFKIPV3tRGDcr5XJHGvpJQj7q6Oho7W1dXh/e///145513cPXqVVy9ehUNDQ146aWX0NjYiOnpaUxNTeGtt95SVFJzbGwM3d3dtGrM5OQkxsbGduQihMNh3Lp1CwcOHIDL5Ypk2aQZgLfQ4wf2iAObqTkB6aTV0dFR1NRVsVM3xa4fDAap01bqSgOVFnpAkrxy1Zzdrd7gtabAKo2BZSEEjHTSyxVKk7WIqFAIXq8Xy8vLAID29va8nVdAHQc2kUjg9u3bWUsvZkJu3EzPFunnLS2tAICEUBn2GMi/qU05ynTVogKbzeZrNBr09PRgfHwcb775Jt5991289957CAQCtALGwMAABgYGMDExgcnJyZzjXbp0CS+//DL9e3BwEM8+++wOB9ZkMsFgMOD8+fP41re+9SUAHgCbACIAtpCMf20F8HMA5go4dEpNO7Byb/tbW1twu90AAIfDQVsIFkolxMCurq7CZDKhu7tbcRB9pTmiaiFXX48oAdLe4NLjL5WxKee5rbQYWKXLMfYG6c4rUV0JhdhBURSxtLSE9fV1mEwmhEKhgl9Oi7GHZAZMEISMlQaUjJ0vyRa0GmxuhWA3mxAJF7SZkqJEYACSIszm5mbJBIa95sASBEGA3W7Hpz71KXzlK18peCy3272j8yGpzT09PZ1Sj9tkMuHRRx/FuXPnAOA3ATQhmcTVDIC8rbwH4DmXyzXB8zzncrkKeljWrAMrZzA9Hg8WFxfpG/Lm5iZdthgFtthOXIWsT8qaAMkLpq+vTxVnoZYc2vQ4LWlv8HA4ae1v3bpV8t7gtWY0lYzFHFgGQUlzgnwdWGlb7MbGRjgcDszMzJR9NkxazYbjOPT29pZtapwoaeubW7CbTUWp2OVCrqnN4uIiEokEVdGlTW2kYkM1UGkOrFp22O12Z2zbTfwoKTqdDk8//TSefvrpD/M8bwTQg2T1gS0ASy6Xa4ssW6jzCtSoA0sMpvRmXlxchNfrTWlOoIYKqdFoyq7ASisNAEBTU1PBF2gtOazZkMYINTY2YmVlBRsbG6ivry9Zb/BafetXMtYuty9kVACkXKGS5gT5OF/RaBSzs7OIRCJob29HU1MTfSFVuyRhNuLxOGZnZxEKhVBfX49QKFRwCbBc48qp1cSB7e9IxsFWgwMrhdScJUJDQ0MDnS0jNWe1Wm1KIlixz7lassWCIJTNDvt8PvqMlEJUWLlxE4kEDhw4oHe5XGEALun3240NinJegRp0YNObE5CyKsFgcEdzAnIzSDNh86XcIQRS493a2krroxYybjHrVbvjS6Yw29vb6RRgeis/tWvOlordMJpK3vyVLMeoTTIla2VCqR2UtmOVJmuplYSlFFI3nFQaCAaD1IkuF+nlHqt1toNcH0RgIE1tSCIYERhIOII0ESyfMYDacmDJeVOyP8VeG5nU1/SwAoJGo4FGo0GmRgUul0vkeX4/z/NdLpfrfxa6XzXjwBKDGY/HqfMaDocxNzeXsZyJGs6YGiEESsdPrzRQV1eHlZUVVUq/7GVIkleu3uD51pyttUoHgPJwm3LXHmRUDunlCpV01lJiRz0eDxYWFmAwGHa0Y1XLgVWyvlylgVAopFoNWaWkOyW1dL/pdDrYbDbYbLYUgSEQCFS0wFCrIQTZSsGlf7ewsIB3330XFosFX/rSlz4GIIxkAhf5f9zlcq0A4JFsbLC3HVhRvNucAEj+WD6fD/Pz81mbE6jlwJZjfZ/PB7fbnVJpIBqNFjUuQ570Vn7SYt2k3IjRaKRxWplqzpbLmJVzuj4fB5aV0Np7SONdifOqhGyhWNJkLYvFAqfTmbEda6kdWOJEp1caUCMZVw65+438zXEcbi6soaetETqtpmoV2FxkExiUNrUptwJbakRRWYtYtarBDAwM0HABoriSf5MELrI/d+7cwblz58i5+B0AegBkRzkAHM/zfgD/GsCfAfjTQver6h3Y9GQtALT3dK7mBNIQgkIptQNLkrWWlpZ2VBpQwwHfqwqsUkdMo9GklISJRCI0Tktac1bN3uCFUikObC0qz4zcpJcrzOf3z6TAlquzYK71RTG1PW2mSgOFzDwU+gxJCAJuLqyh0VqPBms9qtWU53vOCmlqU07HshyhU0qV1WIcWLfbjampKdrS+8iRI7hy5Qr9+8qVKzhy5AhdnhzzwYMH8fnPfx5+vx+vv/76MwAaAWiRdGR1uJvM5QVwM+8dk1DVDqxccwKSmaqkOUGlK7CiKGJxcREejwc2mw1dXV0px1PM/hdzg5VKbah0OI6D0WiE0WhEc3Mz7Q1OYrTSe4OTKc5airuSFs/OtgxQvTF5jPzIN95VDjmbEolEMDc3h0gkkrOzYCkdWPJc8fl8aGhokK0bvlsva7PLXrQ2WNFgrd+TL4xKm9pIu72V8jyVu5xhMSEEbrcbk5OTuHLlCqanpzE2NoYHHniANiqYmprC+Pg4dVhPnTqF8fFxTE5OYnNzE3NzcztqwJLjf9/73geDwQCXy/VGpn3jeT4GIJHpeyVUrQMbj8dp9xWtVkvbqJLMVCVtBCvBgc2EVHlobm5GW1vbnjRQlYyS3uBAMmbOYDCUrDd4rcZdMSofMitBRIRCnFdgZwZ9IBCg9br7+vp2tL9Mh4yp9myatNLAvn370NTUlFMBLsR5z7Yu6WkvXZ7jOGg1Gswte9HqsOCD+zuh0+UXssNxHHQ6HQ3Bq3ZIkpdczVlij2/evJlSaSafRDAllFPpBYpzYJ1OJ44dOybbDhYARkZGqPNKyLQsgexPb28vfWnIUuf1n7HXHFjytr+4uIiVlRXcd999Kc0JpJmpuajUEAJppYFsyoMaCmz6unvFSVbb0MjVnPX5fPB6vdSAkpIwJNRArRjRSnVg98q1tFch1/m7776L+vp62iO9EKR2MFOcaa71yT6psQ9AaqWB7u7ujJnYao2fCTnng9iOmcV12C13Oy/m8zyqMxqh3w6vi0QiReVUFEOpVEtpzVlSNtFms+1oakNsshoCQ6UpsLs1GyZN7MpUKsvlcq0XO05VObDSeFfyw3k8HiwvL6Ourg7d3d0pmam5qMQqBNJKA7mc8VIazb1AqQwNxyVrztpsNni9XqqeS8MNgGScljQRrFAqzYFlIQS1T3q5wmJtELGjCwsL8Hg8GZO1sq2vpphAKg1oNBpaaaDUSO8tn8+HhYUFWCwW2Qxwcl5uLa3DIFFelZ4D0rTli9/4M/zqzz+MTzzQR0NAahWNRoO2tjb64kUcWY/HA4/Ho5rAUK5QLoB1RKwaBza9OQH5QZaXl1OaE+RDJYQQSNeXqzSgBDUVWKXr1rKhUxuSCEZKwoTDYZoIRkrCkN7gFosl75Iw5Y6BzXWfMQe2dhFF+eYEahTQTyQS8Hg8WZO1sqGGXRJFsewKsPQ4DQYDBFFEY1MTQqGQ7P5Fo1FYLBYkBAFzy14EQhH6vdJmBhqNBhuBEP7+n9/DFz75AQhicmwSllcuyp1cBexsapNIJKgzW2xTm0pTYJkDWwGkZ7cmEgnaRjVTUL0S1AghUKMTFwCsrq5ieXkZJpMJPT09ivpBsynawimH4ZQzMtI4rZaWFhqnFQgEaG9wjUaTYjxzXQvlVmCVvvWz67O2kJYrlCZrFWsDI5EIdRo6OzvR0NBQ0HbUcGCDwSC8Xm/eCnAxkPuEtE3Vbv/d09ODYDCYsqwgitjc3ITFYsHSug/ReAJLHv+ObWWDxL06LCa8fOZX8UBfR3LbnK7sDixhN22FVquVrTlbiMBQaQ7sbosJPM9/AIDX5XLNyXzXDGAzU7MDJVS8A0umqojBlDYnAJAxqF4JaimwZBvFXLjLy8uylQaUjp0vLPygMqoDyPUGJw4taRVsNBqpMyvXG5yFEDDKQTweRywWo06rVNHKVwTQ6/WIxWJ0qp5cww6Ho+D9K8aBFQQBgiAgEomgsbER7e3ted1PxdpTp9NJ75cN/xYc1mTSGgmJSwgCtBoNRHE7vpDj8M7cMl0/HI3BaNDThOZs6HQ6ur/3drYAAJa9PrQ12OjvUmsofT6TKjKk5iy5Rre2thQJDJXmwFaAArsfwIM8z/83l8t1i3zI8/xHAJwE8A0A04VuvOIdWPIDSZsTaLVatLW1YWlpKafhzGbUdtuBTSQS2NjYAIBdnTZjVAakN7jFYkFraysikQhVAkhvcLmas5XmwFaA0WSUgHTHlUBmxZSi1Wqh0WgQDAZx+/Zt1NXVwWq1Ym1trSgHoFBbGIvFqBNtNBrzdl7J2EBhzxJBENDa2gpBEPHq9E184B4nAqEwLCYjSP136XaNRiO0Gg3am2zQaDgIggi97m6ZqNu3b1O1UC4xiWTeP/ffr+Lmwhr+9NT/gnhCxO2ldXQ22crqwO5GCEE+6PV6NDQ00KY2uQQG0mq71ChtXkOW242mMoIgwOVyXeZ5XgDwJZ7nL7hcrjs8z38cwNMAnnG5XO5ixqgKB5bjONqcgBTzJ1MduRxYUsRYLsNSrSoEQP43Iqk0QParpaWlrNMo2fZbyX5Uu9NcjjflYhxLjrtbc5b0BifObHrNWWKcymU4mQO7NyG2OB2lcZcEcn00NDTQ/AWv15vyXaH7l+/60koDOp0uY1c9pRTyHCAqq0bDYWZhFQfv68U7txfxwP4uaLXJe4ijjYzubn+gtx2PH/oo/vTKP2LZ60dHkx2JRAI6nY52qCKJSRaLBfX19Slq4dT0LXz6I++HRsOhs9mOxfVN6PV6RCIRehzlsvPVEG6kRGAAks5iIBAoaVMbpTZ2N22xRqMBz/M6l8v1tzzP1wN4jOd5N4DPAvh1NaoQVPwThtRDXV1dhcPhQF9fH/R6Pf1BlNxgmW4OtRVYpQSDQczMzCAWi2Utzq10/HI7k9VgbCoBNZVRUnO2o6MD99xzDzo7O2G322m5LgC4c+cO1tbWiurLngslMbAshKA2yWZHlTqw8XicOqsAaEer3Uio9fv9mJmZgSiK6O/vp7GhhY6dL6FQCDMzM2jYfgaIoojFtU3odTosrW8iITmnxJEFOKqgiqKIX/vcJ6DVaJAQBIiiCL3BgK6uLuzfvx/t7e2or69HIBDAwsICbt68SV92BVHET+ZW8KF7uwAAW+Eo2puS1Q6U5F9UG2oLFkRgaGpqQnd3N/r7+9HW1kZnI8j5np+fx8bGhuqqdr4hBLv1zHa5XHGe5zUul+svAQwA+C8A/nc1nFegChxYUUwWyt63bx86OzvpQ5H8ILuhnhazjc3NTdy6dQscx6VUGlAji7cQql1J3YtwHAez2YzW1lb09vbSlyCtVguPxwO3242bN29icXERPp8vr+ndbIhieftvM6oDpUlckUgEMzMzKYlC0hAxoHh7rnR9j8eD2dlZGAwG9Pf308ScYh1Ypev7/X7cunULFosFep0OMwuryRkVUzLbXa/XQavRIBpLbTDAcXfP1fj3rqOrxYGnH/04nC0N4DgOum0HlXSoam9vx/79+9HV1QW73U7V3tuL69gKR3FPZzPmV734y2v/RMeoVQe2lBCBQa/Xo76+ngoMsVgMKysruHXrFm7fvo3V1VVVBIYqioGFy+USeJ7/ZQA+JB3YIzzPZ+9MopCKf8LodDrcc889aHi2kZ8AACAASURBVG5uTvmx8lFgs5Hv9Fc6Sg2XKIpYW1uD2+2G0WjE/v37VSmcXKjRzTbuXnBqKz2EQCkcx1FVpbOzkyovZrMZwWAQS0tLuHnzJtxuNzweT8r0YDFjZmO33/oZ5UWJ4+j3+3Hz5k0kEgnZWSc1xAQljrQoJttzkxqrfX19tERSuRxYr9eb4jwDwH+euAYAaLIln+smgx4cxyEQjKScWzLOyoYf/+EvriGeEPDs//rprOOSUKPW1laIIhBPCPjnn96h471xcwHP/fdJxOLbL7pc+dyCctqKclZpkQoMvb29aGlpgVarhdfrVUVgqJY6sDzP1/M8/78B+JjL5Trpcrn+EIANwC/yPF948fNtquJVS84wqfHGDqhTxxXIbrhEUcTCwgK8Xu+OSgPFGu5ijW76ukpv8r3g5BZLuYyzdBxScza9N3ggEKC9wUlJGLPZDJPJpNi45Ws0dyNxgFF+iH2WeykURRHr6+tYWlqC0WhEd3e3bE3NcoQQCIIAt9sNv9+fsdJAKe2aKIo0l8NsNqO7u5uOH9uuqtOwXX2g3pg8R8FIBA7r3SYKGo5DQhDwk9kVfPPXh5EQBOi0GoSjMZjqsjviydA7DhpweHNmAU22ehgNOty4tYhwNI7pW4v44L1d0Gk1NMa+XFn1paac1QGk43AcB4PBAIPBgIaGBiQSCZoItrW1Bb8/WQLNZDKl1JxVSyRQmuxVQn4OwP0ul+tpANgOJ/h9nue/DuALPM//ZaZOXUqoCgc2m6Ol5I0724+nlgObyZFOJBKYm5vD1tYWWlpa0NrauuMCJ/tZKLUWA8uc4/zIVmUjU29wUhKGhCMoqTlbRUaTUQKU5BJIlxEEAYuLi/B6vTmbzZQ6hCAWi2F2dhbhcBjt7e1oamrKa30lYwOZ70VRFHHnzh1sbGzA4XCkhMMBwPu69yEai6O1wYZINEYd2FhCgEaTet61Gg0+/kA//VwURQTDUei0Wuh12ozPNGmXytffm8e/+1f3g+M43Li1AAD4r1dexTfvfYwuOzeXLN3p9/uh0+lgsViK6hi4F8jlb5DQjnSBYWtrKy+BIZ8QgkzJl2XiPZfL9R2e5zmXyyW6XC4BAFwu1zd4nv93xTivQJU4sHIoNXikc1CmUIFijJZ0P+QMBqk0EIlEMhboVkOBLYZKdRbL4STXQghBPuNkqjlLFFog+eAi7W3Ta87mG3fFFNi9gZwNjMfjmJubQzAY3PHinq3ySSkUWFJpIJFIoLu7GzabLeP6hUKeMXLjk0TkQCCQci6k98eD+zsxt+xBe5Mdqxt+1BmSiVoayT4ltiuAaDgOXn8QDqsJ/+OVN/H5T3wQTXYLVXEzHYcoJmNoAeDRhw/gg/s7AQD/sh1OYN52mkVRhGP7WUXKixHnymAwZC3RlS/ltJG7ocBmQ6nAUF9fT20yERjyscW7HP/69vb/d9wYLpfre8Vuv2od2HwUWOny6ajVSSt9G8FgELOzsxBFEb29vbBYLHmtn8/4aju/TDmrTvL53aQlYUQxWWaOFOwmJWHSS/BUS+Yro7xIxQStVptSmqqrqytrcwIiLJTKgfX7/XC73dBoNOjr64PJZMprfaU4HA7ce++9MJvNiMfjVBSRKr8dHR2p8b8ch395z42OZgf2d7Tip3dWMHhgP6ZvL+B9Xa3w+ALUgb2z6kVnSwNEUcRP5pbxxdMX8NqfnMJH398LIHnu9dtVFOTuO5PJBBEiRDF5jn/l8Meg0XCIRGNY928BAIY+wiOREKDVaqDX6aj6Z7PZ0NDQQO1Deoku8l8hjpJOp0NbWxvsdnvR+SjZKJcjV4yjnE1g2NpK/kZ1dXUwm82IRqOK7HCthIBkoqodWCXqqTSIWS5YuhQxsJubm5ifn4dOp0Nvb2/K1I2S9fMdX00VlfQ5z7bPDGVUi7rAcXd7g5Oas8FgkBbr9vl8dBkAOZMOdrN4NqP8SG1YusNYX1+/Y3k5e6VWCIF02+vr61hcXITRaERPTw8tP6V0fSVInSIyY0EcMany29PTA6vVSpfV6XTQcByuT89goK8DH77XiR+++R4Mei2W1jfxQH8nfjq/gvYmO7ZCERgNeoiiiFA0Bt7Zisf+9Qfxo5/M4qH7e1OcM1EUYTAYaKdKANtlyjSIxOKo0+twYfIf8cd/+0O8/sIp1Bn0+Kc/OYV/+9Xn8ckH92NuxYu+9ia6jwRpQX8Sx5luH0wmE1UKc51rvV4Pnd4AnVaTcl5i8Thi0ahqlVPKjVoOo5zAQGbKPB4PXW5paSnjC8RuJ3CVg6pwYLMphUpCCLJtQ80qBGSqZXl5GSaTCT09PYpLkuzGVH66wY5Go7h9+zai0WjWqSK1nebdoNZCCNQcQ6fTyfYGJwkHq6ur2NzcTInTkgs1qGXDuRfJZkMBUOVeqcMo3aaaCqwoilhaWsL6+nrO2Fu59fNB2kREmphrMBhojdl05Vej0dD1Hhroxz/9ZBYPH9iPZY8POq0WXn8Qep0WC2sb2N/Zgp/Or2CgvxPLXj8+9RvfxPd+7yv4P37x55AQRIgA3vzpPA70d2BzK3y3ioHJhFAoRP8diydw4cqr+Pef+yR+9M4sDNuxsqsbAXQ02XHjwm9Dr9PhT1+awqMf/wA+dG/mc5Yex0nsQyAQwMrKCoDsoUhGo5FeG7/7F9+HBsBvf+nn8Df/8AY+fG8XetoakUjEEQ6H8/otslGJIQRKkQoMjY2NSCQSuHPnDiKRyI4XCGki2F6YCasKBzYT+Uz/Z3OC1UriIpUG7Hb7jiD9TBRbDkwtZzIUCmF2dhaCIKCxsRHhcDhjNxdGZVFK40xisEjv7/n5eVrfkFwfpGwMaW9L1BPmwO4t1tfXd1RZUYpaDqwgCJibm4Pf78+7PXchtlQUAREiNEi93jmOg8PhkE18kjr2DosJNxdWodVqUKdPOpUmgw46rRa+rSD0Oh2a7BZoNRq4l71Y3Qzg7Ld/gP/8a78AHYBgOILe9iZc/dHb+PTB+/Fj1yw+eI8TOp2OdoLiOA6/82eT+DcfuAcAMPX2bTz+6YMAgCfO/BU+8cB+nBz5WQDAhSv/CEEEfobvhlambbDcOSP2obm5OUUpTG9/bbFY0NzcDJ1Oh2Akgvq6OnzigX4MDvQCAP74b38IvrsN3/rqF6DRJBVnac3gYiiH4FIuUUer1UKv1yORSKC3t5e+QGxtbWF1dRWrq6uIxWK4dOkSHA4HHnroIRw4cKAs+1ZuqvoJU2wCFtmGGg7sysoKvF4vWlpaCjLgxYxfrPMbCARw69YtAEDvdmF80s1l3759MJlMKd1c4vE4YrFY1U7zlItqVWCzjQOAOimk5qzFYqE1Z3/pl34JX/rSl/D888/je9/7Xsp0l5SpqSkcP34855hTU1MYGxuD251smT0xMYHJyUn1DopRFPF4nKpuDocDTqczL9tHrls1QghEUUQikYDf70d7e7tsmaxc+5K3A7sdV5q+HQBoamra4byS4yRdtvY12TG3nOxM1mhP5knUG5OhOga9HnqdFvsakx2yDt7Xgxd+84voak0mWAWCETz/N/8TdrMJb928A71Oi/nVDWi1GtxaXE9RUM9/7zo+dG8Xbi6sYcXrx89+6H2IxRP45/fmceHKq7ST19Of+wS+88qbELYPKlPuhhxEKWxsbEzpTmU0GuHz+WgSkiiKqN8OR3ro/l56Tr5/5ml866tfAAC89OrbtPyUWtTKjBtwt6U3eYFoaWlBr6Tm7OLiIr73ve/hL/7iL/Brv/ZrOHToEP7oj/5ox/01Pj6OyclJTE5OYnx8POuYlWiLq9qBLTYBi2yjGKNJYo1IpYG2tra8jSawe9UASKUEvV5PmytIs8htNhs6OjrQ39+Pzs5OOq0cDodpgXyv14toNLor+18oLIQgf9LrwJKpxH379qG/vx9OpxMPPvgg4vE4Xn31VXz961/Hww8/jGvXrtFtTE9PY3x8HC+++CINSciGz+fD+fPnMTQ0BJ7nMTc3h8OHD5fmABk5kV5nxAaQe99ut+d9HaoVQhAKhWhL5Z6eHtkyWUr2Ren4Go0Ger1+O7OfzMKl3u9arXZHCJl2+2+tRoM3b96B0aCn5bAat2vAkmoAdWRZrQZP/sFf4dqPXfjsww/gN48k1dJ6ox6v/eQ2OI7Dgf2d2ApHcH9vO6ZvL6KjyYZE4m5y3Kt//BtotNbj//2X9wAA93a14PX35hGOxvHMFz8FrUaDlY0AnvnFIdzT0YxX3ppBIiFkrNigBGn76/e///2wWq1ICAICoSgSCQHf/7ELnUdGEYsl8N78Ct6aWaDr/rf/50eY+PvXVXNiy5nMVC4xQe5FkdSbHRoawvnz5/G5z30OfX19uH37Nr71rW/RewQAxsbGYLPZcPjwYRw+fBhOpxNjY2MZx6xEW1wVDmyp4lfJtgs1msFgkL6NNDc3y5bJUjI+sDshBIIgYGtrCyaTCf39/Vlj1sg0cVtbG/R6PUwmE43HWV1dxe3bt3H79m2sra2p0iqPoZxyK7ByY5EYrK997Wu4ePEiTp06hZGREdx3330pYScDAwM4duwYHn74YcXjXrt2DdeuXYPL5cKpU6eKPxBG0fh8Phrj2dHRAaC8LbmlkLasBGlSUL4oHZ+UzeI43K3Hip3NdqQqqFarpVUFBFHEH1x6GQBwcLuSgMOSvE/MpqQ6aazTIyEImFlYw3deeQu/Pf4dqvb++//yIjQaDR77Nx/CzYVVDA7047tTN9DX3oxz/+OHqDPoodVqcPT3/hzP/80P0N3WCIDD4EAfPv0RHhZTHf6/t2YAAI9+/EHcXlrHz3/tTxAMx/Di//UrGOhphVarQVNzc/4nMQ2iEgqiiIW1TXz5P/05tFoNpm7M4J7OZtQZdPjLl3+MTz/zLURjcSQEAX/19V9Gi8NCVd1qiOMspwKrxObzPI8jR47gm9/8Jn7wgx/g7/7u71Iqgly6dAmPPPII/XtwcBCXLl3Kus1Ks8VVHQOrRvxnoduQVhpIJBKKEhYyjQ+Ut5EBSXIQBAF6vR69vb074reywXHJGobNzc20fh3JSPV4PPB4PNBqtTSIn8Ri7TVqTYFVejwGgwEf+tCH8Cu/8iuq/O52u70oJYihHqIoYnV1lSaqdnd3UxFBjXra+W5DWmnAbDZjfX294PuhkHU0nEZSqzW1oYM0PEIQBJAWrYIgwr3iwT+88VMAwIG+Dvi2Qmiw1SMWT9AQgp95fw+0Gg1aG6yw1dehwWqGTpvcxo9dc5hd9uDTP3Mf/vz7P8Kv/8K/xatvz2DkZz+Cpz77cQDJMIUrP3ob371+A4c+eh94Zxvu79mHP/8/fxmiKOLLnz6Ij77fCYfFhHdmF/Hd//AV2MxGCKIIo8FKz8n+/fuRSCQKFiZIGAIH4G9++AY+9ZH3AQBefWcWv/CJDyaP5715tDos0Ou0GP/uFIY/+QH87IffR7dRV2dEOBzKe2xCLc24kbFy2VbpjFl7e3vKd263Gz6fL8Wu2mw2+Hw+TE9PY2BgQHablWaLq9qrUKrAZruw8lVxiQF3u9200oB0jHxRQ4HNB0EQMD8/j/X1dWg0mrxaiWaClFhxOp0pcbM+n4/GzS4sLBTc97kUlNOglZpKc2DV7sRFYq0mJiayTnExSguxHcvLy7Db7ejr64Nery/KhqVfI0q3IYrJ9tyLi4uwWq3o6+vbUeS9kH3Jv4zW3f2Xqqvp29Xr9dT51Gg49OxrQiAUwZ21DdzrbMXcsgfNdgtWN/ywmOoQTyRgNhkRiydgMdXh9XO/hdOPf4bGzv6nJz+H//sf/gUNNjPemrkDDcfhZ/huhCJR3N/bAVEUodVo8B+f+HkAQN++Jlx97R189FfP0H2ymo34xIPJxK6HBvrR2mBFIiHQdrWbgSA0HAdBEFNqvuaDNIaW4zhc+cd38LMfvBfBcBRv3LyDTz64H/GEgH/56Twe++QHwHEcXv7n9/DRr/wBtSP/+PZt6PU6WuKxEGrNgZXWTs61P3LPd7fbndER3dzczLjNSrPFVe3AKjU42R6o+RgtYjSJAe/t7aXxObvpwCpdN5FIYHZ2Fpubm2hra9uuD1jYzZZpTGnc7P79+2ncbCgUwtLSUkrcbKHGqNqoFQU2PQY21/6osU8DAwMYGRnB4cOHMTIygu7ubkXJX4zSEIvF0NrampKoqlYNV6W2jLTn9ng8aGpqQnd3d4otK9aWKllfvpYtGf/uZ4IgJmu+ypyjLx/6V7gxs4Bmuxm3lzywmOqwsLYBu8X0/7N33uFRVG0b/83Mtmw2PYFU0ui9i4iIFEWxYEURC4oNff2siL6IvaK+WFGwIIoUG00FRFHpvXfSe+9td2fm+2N2hw0kIaFJkPu6uLiyOztzZubMOfc8537uRyfD0xat5u63vsXHaqFfp1j+3H6Iv3Ycon/nOH5evxtRELisT0cy80sYdWlvvMwmFEVh2JPvszMhgzHD+vLI9ZdgMhpYvHY3qbnFOJwyC9fsYviEaSiqqpNiRVGodjj4eMHfZOYXY7Nq0djk7AIU13VxS8kaY0tmMplQVJVN+1PJLyknI7+YbYfTiQ0LYv3eZJyyQtuoEHYnZVJtdzKkZzsUVWXzgVQUjoybfdq30trj40NeXh4pKSnk5+dTXV19VknVzrYIbEMEtrS0tJYe1g13FLYunI1jcbMgsCergW1o4m3KoJmcnHyM08CpiqCebgLrcDhITEyksrKSyMhIQkJCTjgJrrEPqKdu1p3k46mbTUpK0nWzZ9tgdCrQnL0H6zsOnNnyhVFRUbUiBf3792fZsmX1DrLncfogiiJxcXG1ysLCqUtEbcx47nA4SEpKqtNp4EytZlVWVja4kuTei6KqiKKgJ3fJsqJ/qygKj9wwiH0p2ZhNRvKLyzAZDWTml2DzsmhjM/DV0g0sWbeb8irNTmr9niS+WroBk9HAsF7tSczMZ2iv9mw7pBWPkBUFURT5z3WDeOzD75FEgWfHXI7d4eTXjXsZM6QXRoPE0o37+PWN+xEFQW+vKIpYjEZah4fgZTYhiZo8IizIj9e+Wea6P5rS12S2YLPZ6i14I4oiZrMZSRTxsZoJ9rMREezPxo8ex2Iy0jEmlDFDe+PjZWHtHk2/3CkmlAOpOZRV1jBqUHcEQeDJaQvIKiylRYAvkijqcrfCwkJSU1NJSkoiJyeHioqKBvvOuRaBbcz5NBS4qy/6erSswBNn41jcLAhsfWiqjVZDEdiGBj273a6Tv/qcBv4pAtsYVFdXk5iYiMPhIDo6usHSjqcL7iSf4ODgWnYfkiTpg1FiYiI5OTk4HI7TTmbPZFbq6cbZRGDdz9KpaE9paSnt2rXTEyVB02ABtT47jzOH+mRYcHIRWPe+G3ruq6qqdNeDupwGzkQwobKykoCAgAYL1AhuWYGqMuf3zYiiwPTFa5AkkcSsfBQXyYwM8aeoTFumD/bXvF5FEYwGSUuwrbKTllfM5X3aY3Mldj1282D+2HqA/JJy7ryiHz/8vY0AHysjLuyMispvm/exMyGDy/p0YGdiJm/N+Q2AarsTh1Pmriv6oSgqUx+6DqPRgN3hJDlbs7pbunEvG/Ylc1nfDgT6WPn+r22MnDQdVVV5ZvRlmrxA1Ahvtd2BompVv6ze3lgsFl2H7OPjg7e3t34d27dqCUBucRkxripfoYG+vDv+OkRRoFNMGEN7tcPP5qUnlo0c0BVZUViweidjXp2FqqooioLZbCYuLk6XqrktujIyMnSpWklJSa1KZO57+m8lsHUFE9zjaF2o67uzdSxu1gS2sRHEhkLpxyskUFlZqXufxsTEHOM00JSlr9OB4x27oqKCpKQkVFWrCHO0Jumfarfb7qMu3Wx5eTnAWaebbSrOxQjsyeiuTgTjxo0jKipK/9s9WNaXZHAeZx5HbKROH4EtLS0lKSkJQRCIi4ur02ngdBJYd+Kr21i/of7t+QwsWbsLgHZRLQBwyvKR3woCz92pZYG7vV6v6NdF/62P1cy+r/7Lmw9cS1WNgyc//hGr2cSUB6/n6+UbaeHvQ882UXogp7LKzgUdYvlk0Sq8zCYm3DqU2LBgJFHE19tCwrfP0zk2DEFAcykQRbIKSolqqc1pD747jylzf9dXUK4b0I3OsWF8+NNfWuBYgM9/Xus6D4WDaZr/r+RyWxBF6YjPraxwIC2XoU98QFWNgz3JWXQe+zrZhaVkF5bS98G3KSqrBGBAlzi+/e8dWoQ1LAiDQaRLXDg7DmdQWllNSnYhgnCk0pnBaNSrBR4tVauuriYnJ4fExERdZmK328/IPHemNbAnIyHo1KnTMXIBd/S1vrH1bByLmz2BbUoSV11oaNAqKSkhKSkJSZKIi4urV8B+MkTwdEoISktLSU5O1tvvWc7wbMLRulk3yW7uutmziVieCjQlceBE25OWlsa8efMAbZnr6NWCefPmMW7cuBPa93mcPBqSc52sHKmu8VxVtfLcqampmEwm4uLisFgsDe7rVI/F7uS1/Pz841YiVBQFSRS55cUvALh+YHdScgppG9WChIw8gny9KS6vJL+kHFEQMLi0pL3ba8nA+1NysDuclFZU8dwXP2M1Gwn192XVrsPMWr6RDXuTGTmgKyu3HUQQBIb0ao+qqtgdMtMWrcLf5kVmfgkZecWMv3YgIwd0Y1diJqNe/BKjR9lbt6whOjRQ/3z75xOZ9ewYKmvsDH/qYxKz8nlp7AieGKX5xEqiyOV9O/LMjEX427yIDQvSCyCIosjG/Sn6dZAkkTe//Y12US3xMhuZtWwjvlYLIf42lm3aR3J2IQZJZMvBNNrf8Qo5RRqRGtarHcnfPo+3xcTyzfsB+L8bBmGQRHYnZaKqKgZJquUN6ylVi42NpVWrVgQFBel9Jzk5GdDmk8rKytNGZs+2COzxggk333wzv/76q/73r7/+ys0336z/3RzG4mZBYOu7UU0R3R9v354Dp6qq5Obm6k4DcXFx9Wp9PNtxMsc/1Q9VQUEBqampWCwW4uLi6jWDPtFJ53QNAm6DcHekJSoqioCAgFq62VMh4j/XJARn6jgn89YP2qA4Y8YM5s2bp1d28azmsnbt2loVYe699159+ylTphAVFXVW+A+eR200Vc51dKa+ex+efVlVVbKyssjOzsbHx+e4XtUnGwmuayx25z64E1+Pl4UviiKKqrInOZukrAIu6d6GLQdSCfL1Zv3eZAJ8rGw5kEaAy/NV9mir3eHk1W+WYjIamLZoDZ8uXsPhjHxEUWBIj3aMH3kxz8xYhNEgsei1B/TfvTVnORazkaoaO7Ki8uLYEbz41S94e5kxmwy8Pe93Vm47iKJoy/CyolBQWsF1z32G0ymzOymTF2b+gqyo2LwsVFY7SMjM49r/TtdWGBF474eVvPjVL0SG+HN5n44AmI0GJFEkMasAgNLKanYlZujt+nzCaKY+fD3Vdjvf/72dB665CEkU+WXDXqJbBmDzMvPn9kOUVFRjNZvYdiidu9+cTZXdgSAIjBvRn15tIxk1uCf5JeVc+9/pJGUVIMsKJpOpzkQyQRCwWCwEBQURHR1NbGwsISEhgCalS09PJyEhgaysLMrKyk7p6t6ZIrBNyUWA+sfip556itLSUt1VIDU1tdbY2hzG4mbtA+u5/H+ineZoCYGiKGRmZlJcXIyfnx8RERHHnbTPpgisqqrk5OSQn5+Pj49Pg6Udj6dl/KcJnls36+XlRUhICHa7nfLycioqKnS/Wc8621ar9R9vsyfOtQjsySYOgJYIcO+993LvvffW+f2oUaMYNWpUrc/q2/Y8zh40NQKrqioc1Uc8xzJZlklLS6O8vJygoCBCQ0OP2/eOJwdrUts4UqXQbrcTGRl53NwBWVaQJC3x6YlRg1m18zCjh/Zm++F0rh/YnfySMgySxJBe7XQCL4kin/28lvLKah69aTCzJ92FrCjMXrGZiGA/OsaEsmLLAaJa+DP5jiv1SCVAclYBMWFBXNAxlorqGu6+sj8LVu/gugHdSM8rosbhxGw0cOvgXnz59G0ulwTtGu5JzuLxGwdhMEi8PGspf+1M4KlbhlBeVUOIv41tn01k68FUl0QOLZr73HR+XLWD6y/uBrheWlSVmUvX06ttK0YO6ArALxv28OrXy5h853CG9e6A0WBg48dP4OvtRWllNWt2JfL8nVcgCAIrthwgLiwIm5eZFVsOsGT9Ht647xrwBm8vE0tefwBBEPhk4SrKKmsYN+VbVrzzH1SX/raqqmFvWKPRiL+/P3l5eQQGBmI2m6moqKC8vFyvBGi1WvU55ET93OHsI7CNcY1paGxtDmNxs4jA1ocTIX9H33TPfbhtpoqLi2s5DTRmnycbBT4VBFZVVTIyMsjPzycgIIBWrVo1OmLWHGAymQgMDCQqKkqvs202m2uJ+LOyss4a3ezZRCzP1HFOtQb2PJoHTkVVRPc+7HY7SUlJlJeXEx4eXstpoCGcimCAOzmrqqqqzsTXhtohSSI3Tv6MjPxirurXmW2H0jFIku7/+n83aiVgFUXzWv1j635Scwq58ZIevPrNMiZ/vhhFVRGAV+4ZwY8v3YMkigzt1Y52UVoiVGjgkSzw0CBfdhxOZ0CXeJas3U1kiD+bXcv4v7z5EEZJJCu/mOEXdMRNXN0R30Hd29C/SxyyrDDhlqF8/OhNeFvMPDN9EXe8NguDKDCgS2u2HEih/0PvUGV38P0L44gPP5I4d/MLX5CYmc/zd17BjsPpOinvEhtORn4JLQN8QVX54Me/kBXNkcFqNvLTy+O4ZXBPCksr2HY4nfHXXowgCCzfvJ/QQB+C/byZvmQtAx7+ny61uG1oH/p2aMUl3VpriWSue9VQMp0bnmOSu/R1fHy8vrrndDpPyeqee3tJkjAajZhMJsxm8ymvJNZYn+1/w1jcrM+sKdmv9b21uPdRU1NzXKeB+tDU5bO6fn+yRNKTfLdo0YLw8PDjtv9silY2O1gNcAAAIABJREFUFe462xEREcTHxxMeHo7NZqOyslLXzaanp9ermz0bIsynEmeLBraxXrHn0TxxPDnXye5blmUSExN1p4HAwMAmt+1E2xEcHEyXLl3x8fHBx8dH19x6Jr7WB1VVccoyf+88zE+rdhDo501ytra03jk2vNa2hzPy2J+STfc2UXz28xr8bV7cNKgHkS0CXU4EIldf2JnYsKPLuGpjlpuEmgwS3VpHYpAkLu7amsoaOy+MHeFxHSAs2B9VVUnNLdT3cvMLX7BhXzKSKLJhfwrd20Qy8qKuOJwyq3YlsHTjPsqr7YiiQNf4SCJC/Bn7xtd4mY10jYtg0/4UKqvtfPH0bdw35VtUFSbdMRxZVvj+z21EtQhgx+cT6RIXznd/buflWUtZvG43kihSVeOgT/torGYTfjYvfnppHDdf2oPc4jJ2Jmby5M1DEEWRBat3kFdSToDNSmJmPgaDyMJX7+e5O4aTnF3A+r1JKIpywnkd7tW9kJAQ3RUnODi4lkWX2xWnvLy8UfO7JEnExsYSGBiIxWJBkAyYTCbNqcHq3aAMsSloLDH9N4zFzeLMGho0oXEEtr63Fvff6enp9ToNHA8nmsDg2YaTiRooilIrYnG0T2NDONs0sCcCURSx2WyEhobW0s06HI5ab9YFBQVn1G/2bIqMnqrjnKwG9jzOTTQ1AnvEffQInE4nAQEBdO3atV6ngQb36YrKncizIEmSq+CAQJXdjiRJtGvXvlbCmMlkalCmYJAkHrjmYtbsTkQUBLrGRVDjcHL9wO4oikKN3cHKbQeIDAlg5bYD+Ht7sWTtTsoqq5l0+3BGXdqTrIISZNeL4v7UnFrHcvu85hSU6n6vbjIbGuiLt8WMxWTUCypIrsivIAhkFWjVlb5dsZm1e5Lp3jqSnQkZjJw0ndm/bdJyDwwSmz55igWv3Eewn41vV2ymrLKabyfdxeihvfV9SaLIA+/OxeZlZuYzt+ufCcAzMxby0NR5+HiZQYXErHwAbhjYnfS8YlqPeZmvlm5AkrTtL+wUi9EgEeTjzaZpTzDq0h4AbDmYzgt3XYnJaGDylz8z8JGpOrkPC/LjrW9XUFJRjeyy1moIjVly91zdq6+aZEZGRp0WXQAWi4WQkBAsVhuyorB6VyJRt7xAfkk5Tllh7sqtOpE9WZwqDey5gGZ9Zk3RPNV30ysqKvR9NeQ00BD+SQmBoija4FhTc8IRi3MJnm/WsbGx+pu1IAh6YltSUhKyLON0Ok8rmT1T1i1nE1E+1WVkz6N5oKmrUJ7dw50tXl1djZeXl56IcyJtaN269Qn5XLsTghRVZU9iJqBV1vLURHr2aXcWv6KoKKpKTlEZhaUVjB95MVsOpAIwtHc78kvKUVWV0ooqRFFky4FUvMxGdiakI4oio4f25ZOFqwgP9sfP5kXLAB8EBGRZIa+4TCezsqzwxMc/aG0QBDLyijXNrV4NrbZEQGubQnlVDXaHkws6xgIwemhvDn49CbPRwKzlGwG48ZIeJGblc9Uzn5CeW0S/jtHYHU4mf/4zIyfNwGiQuG5Ad0rKq5i3cis920YxpFc7Zi5dT6uWgRgkkfl/bkWURFZ/+Dh+3lp5ckkSeXr0MFZ/+Bj+Ni++XLoeVVW5rE97CksriLhpMgtW70ASRX7fdpBWLQMxGTVJQMZ3LzF2+AUA9GwTxYRbhgJQ43AiAN+9dA+iKCAKQr0JXW40VZtaXzXJmpqaOi26zBYLoiixKymLcW/PRRJFBnSJY8KowfjbrPy0aiePfvQTe5KzkSQRi+Xk3IDOE9gjaNZndjISArfTQH6+9oYYGhp6wiH+UxGRPJHfV1ZW6nWLY2NjmxyxqA/nEvlwv1m3atWqlm5WURTdGP10ZKS6caau5dlCYP8Ng+Z5HIuTWYVyOw14ksUT6T9HNKxNexZMJhOCIFBjd6DIChaT1g6HLGvRRUk6JljiPlNFVREFgZlLNzBr+UZaBvgQGxbEofRcBnSJJyLYH0EQeGfeCowGifLKGgRBIDTQl7LKaq4b2J33f/iD7MJi90kwc+l6sgtLuahzPJIocs+bs1FUlcdvGsL/5v9BeLAfJRVVSJJYy80hMTOf7YfT9XaKokjfB6bw5pwViILArGUbmLVsI2aTRhLfvP9a9n31X6wWE//7biUb96eQlleMKIqYjAa2ff408eFBOF0FDLy9TMxatoE1uxK4fVhfbr+sr37sQ2m5PP3pAloG+PLKPVeRW1RG17tfZ94fW2gb2QJZVmgf1ZLOMWG0DPRl3sqtyIrCoO5tSckpZMyrs9h6KA1FVXn2s8XIypGx5vGbB3PToB7IskJ1jQMVLfnNz/sIETyevdmJoiGLrqqqKqzeNkxGI5Ik0i0+gm8n3aH3kSduuhRRFLhpUHdmPj2a2FAtuGQ0Gk5KTtDYIIF7u8aU/W2uaDazTF03qymaJ8+JVVEUMjIyyM3N1fVNJzPh/hMR2LKyMt3YG07sAT7bpACnG566WUmS8PLy0nWzWVlZum62uLj4lPjNnonIaHMyzz6P5o+G5FwnmgdQWFhIcHDwCa1+Hd0G7X/0JJrjQZZlnQQ6ZQWHrODn7UVVjZ3KKnutxC44EuEUXceSRC0Tf87vm5n3x1YEQeDBay9mx+EMFEWlrLIaQK+kFR7sh6KoDO7ZgWUb9xLdMpBAHystA7RCBkVllbw973dMRoMeVe0QE8qb3/5GRIg/3l7aOXWMCdNkBILA5gOprNi8n7jwYEorqpHlI1r1dR8/wYRbhpCWW8Szny0mq7AUSRR574c/mfP7Ft3O66WxI3jlnhEM6BrPknW7ueet2RgNEp8/PQaDJDJx+kKq7U5+fGkcv205oD/nn/28ln0p2UwcfRllVTU6cQ7wsSIIsGjNLlRVpaK6hpsG9eC3tx9CFARKKqoZ3qc9/jYvZi3fSAt/G13jIliybg/f/bkNAc0pYegTH+j7lCQRP5sXZuOR++HZH93aZW9v71ovQ6dqjPS06IqLi6NN27Z4WY4Q0Rq7k9KKagRBoNu4N3n/x78QBYGU7EIu690ek9HAgtU7+fyXdceNGjeExo6x/4axuFmf2YmWMPRMdmrZUsvu/KckAO7fNwVFRUWkpKRgNpubrNdtDI53Lqeb+J4JUi0I2vJgXbrZ3NzcWrrZmpqas5bon23m2f+GxIHzOBYnZKPlQnh4OKGhocfUWm8qjhBYd3XEhrd3OBwUFRV5LMOLlFVW4e9jJTEjD9Gjq7vJhijUXrJXVY3M2h1OEjLzSckp5LLeHbiqf2cEQeDtucsBiG6pmf53iYtgT3ImXePDWbRmBwZJYutnz+qShCBfb3Z98Swh/jZe+Xopu5My+b8bBrF80z6SsvK5d8RFWntEkYNpOcz5fTO92kbx147DbD2YxqDubZAkkWFPfMBT037CIIqYTUYCbFbGXtGP8dcMIKewlLfm/s6BtFxEUeSX9XtwyDL3XTUAgOLyKn5ev5fM/BKdqF/ZrxO3vPg5ZpORyXdcgSRphQ3uuKwvr81eRm5xGR8/OgpBENiXko0gCGyY9iRfTtQKI1w4/l0eeHeuHhyfOHoYs569A0VRuf/qi9jw8RMYJJGLu8axZ+azGA0S7Vu1ZPmUh44Zc9wvEZ7RZ08OIIoiFosFHx8fbDYbPj4+J6yNrguSJB0TMCqrrCF69AuUVVWTkV9CVmEpfdtHI8sKAx99nwmfLkQQ4JqLunAoPY/swlJMphOLwp6XEBxBsz6zE4nAgjZwRUZG0qJFi1NSx/tMRWDdsoeMjAxsNhuxsbEn5X14Nkdgz0Tk0nPC88xIjY6OrqWbTUlJISkpidzc3CZVcjnXIrDnCex51IemJHFVVVXVSoRx6/ZNJpMe7TyR/nxsgm7921ZXV5OYmIi/vz8Op8yWAykYDSJ5RWVYTEYOpeVgNBrIK9a8Qt2PvCjW3qn776VvPQTAso37sJgMmFwJYZ1jI6iqsdM5LpyDqTm0bdWSP7cdxMfqxcZ9KVRU1SBJEoIocPmEj7j/nbn6Pg+k5fDg/7RKSN9OHovNy4IoCpRVVvPlr+to3yqUVTsTSM0tYvKdV9C+1RHnnEduGMQv6/dgMhpIyiqg2uHg5buvws+m6VPDg3x57MZB5JeU88D/5vL6bI1oV9c4GDOsDxnfv0xceDA//rWd7//cxsCurVn46gP6OcuywvMzf6bG4eTzp27D19uCKGgVvj5a8Dc3Pv8ZFpNRuw6CgFNWuG5AVyRRZOyb3zD1+5UoikpaXhEmSXIVYnAQ6OONQZLIKynXk7Y8x1tFVZFEEVlWdHLtlBVU6i6OoagqRqORbt26ER4e3qiofEMwGo06eXXKCmPfnI3d4WTVrgQURSXE34cN+5IBaBsZwoG0XKrsDjrGhCKJIg6nzKvjrqKgpAJJEhtlA3Y0mkpgzyVJ4NFo1rNMU8inO1kLqOXrdyrMr88EgVVVlczMTHJzc/H39yc6Olob+M7hzvlPQBAEzGZznbrZkpKSYyq5NNT3/o0E9t/w1n8ex6KxY2BpaSmJiYnHbHsiE/nROJqc1NdXKyoq9DZ4eXmRkJ7L7sQMQCA9rwijQSKvuByz0cCepEy9QEFdqKqxM23hKqJDA7nv6v7YnU5EUWTT/mSKyirp1a4VOxMyiA4NYs2uBPxtXmzcl4QoCrx49wi8XdKCnQkZbDuUztgr+unt/uyp2ygur+LjBX8TGeJPiL8NQRCwWkwsXL2TkooqXr/3Gh54Zw5Gg4SX2chDU+fz8/rdXN2/C3u+moTRIDH+f/PoMe5NV8InhPjb2PTJUwT4WBFFkYs6x/HiXVeSX1JO93vfYPaKTTo5vKp/ZwJ9vVFVFUnSXA+WbtxLUXklz90+nJyiMiRRxGI0sjc5i8yCEt77z408NWqIyylBwMtsYs9Xz3J5347kFpXy8/o9XHtRF2RF4brnPiPXRVZ/WrVDu6gqhPhp0j53m90oq6jW2zJ90RoURUtiXbRml4tAa+OxU5ZJzTliHeZOhjObzdhsthPSoHp5eWF09TFZUViybg+V1Q5MRgMrtx2iX4cYTAaJDftSEEURX28vvbzuBR2iqbY76Df+HewOmU6xYaiAKElNDp41NkjwbwgmNJsza0gDe7wOUFxcTHJysj5oenrHnax3oHsfp/P3iqKQmppKUVERISEhRERE1IoewrkXgT1bcLTfbFhYGDabjYqKimN0s3XZq5xunGkN7HkC++9GffffHYGtbzxxOw24y1sfr+JRU/uzwWCoZXNVH0pKSkhOTsZgMNCmTRtEUaRtdCjJ2QUYJJH84nIAvC1aOWuHw4koCpSUVx1zbrKisHFfMq9/+xvZhaVMvvNKHrjmYgD8bF6s35tEeLA/G/cl420xsyc5C0kUaRngR1W1nZsv7a3vq3vrSA5+PZk+7Vvx06odXP3MNAySyOLX7sfXlawkKwoXPjgFRVF5e/x1vDjzV/xsXix87X79ml1xQUfueetblqzbhaKo2B1OahxOhvftgCAITJm3giEuXamqQqCPlTnPjcXbYibI15vvXriH24b2QRAEXpz5CyajgUu6tdajm9V2B8P7dsTfFcmNCw/W5ASSSNuolvy6YQ+KotCvUyySKHLzC59z7X8/1WUaLQJ8yf3pdaJbBmI0SGz59CniXH63twzu5TqP2nIQUdQiuKqq4utt0b+775qLUNEisnHhQSiuErkvz1pKdkEpEcH+oEJGXjF/bjsEHHkJNxiNusSgMWOVt7c3gihSXlXD+r1JSKLI3D+2cMflfQBYuf0QN13SHYAN+5IZ1qstBklky8E0AGJDg9h6KJ2M/FImfLpIL1hhNpnIy8tr0KLraJyPwB7BOVNKti6oqkpeXh65ubm1yox63tDGkuCGcDIJDO7f13cOTqeTlJQUqqqqCAsLIygoqM7tzjSaO/E9keiou5KLj4+PnoXqLkuYm5tLbm6u/oZvs9lQFOW0Z4CeqUHqvHn2eTQEzxfpo/uie/WoqKgIX19fIiMj6/39qWrH0VBVlYKCArKzs7FarbRq1eqIVZcKKa6iA15mjVj7+WjLxKIkIggC+1Oy6NMhtpYsQRQEBnZrw/6vJpFZUKJXydqwN4kebaOY/8cWrrigE3lFZYiiQFQLLV/h+oHdMJkM+rg/e8VmDqXnMen2yxGAvSlZbNiXwjvzfmfCrcMYe0UQlTV2rGYTj9x4KZ/9vJYHrhlASYVWRlUSRXYlZJBRUMKV/Tpx/zUXcUGHGGpc5Hv52w/jlGWcssyMJWu5sl9HBEHgtdnLWLsniUWv3EdmYQk1didd444UXXj+ritxyjIGjzHMajahqGqtzwC9ffdedVGtz+dOHqs7Crw993fMJgPjr72YgrIKTAYJf5sVgyS4ErWOXFxZOWIRBrBpfwodWrXE38fKp4vX0MLfxsgBXdl2KB1VVendrhWCIBDo481FXeLwsVqQJJHi8koiQvwJ9LXq+3Q7R7jhTh50O9N4zuXuQgTufrVi8wFGXNgJgH4dYxjcow0AbSKDGdgtnuLyKval5vLQtdqLzMYDqQT5WvG2mFi/NxmA6Jb+tY4fFhbG4cOH9VVii8Wil7b1PLYbTSGwbj34uYpmPcs0RD49nQb8/PyIiYnRv/OcXM9mCYHdbicxMZHq6mqioqLqJK//RAT2XH4gGgtBELBarbV0s0FBQbV0s3a7nZqamibpZpuKM01gz0dgz6Mu1DeOyrJMcnIyRUVFBAcHExUVdUzfqKuvnGh/du/L/Xu3htJt1eXr60tMTIyraIFLgqaqpGZry81uayarWVsqNrmkDdV2B6J4bKBiy8E0LGYjrSNCkBUFh8PJnN+3YDIY9Gz5LvERWvS0UxyJ6bn06xSn79PulHl99m8s3bRX8zUVRSbeehkjB3Tjh7+368cZ+th7bNqfwo2X9ODr5Rsprazms6duAzQCa/My8+C78zickcfkO64gxN+H12cvZ+AjU9mXnI3ZqBHmpW+N5637RwKwfm8y7z54HZIksnD1LloG+NTSnKp1EFVBEGqRLzfc18sNWVHIKizRkmUN2j6evGUID103EARo4e+Dr7W21687kc19Tp59qVt8OP6ul4qebSK52pUkN/qVL/l752EEQWDeH1tZtyeJwT3a4mM1U1RWSd/73+aLX9bp93HltoMsWr1TPz833E4HFouXHpn1dkkNBEHQI9A3XNJdv69P3DwYs2slYc6ku4gM8cff5sU3z4zhpkE9KCytJCmrgFGDeiAIAhv3a/7Al/ftQI3DSXpeMbKi4O3tTVxcnD6HAPockpycfEzuRVPG4nN9HG7WZ1ffoOl0OklOTtadBiIjI2s9mHVFYE+GYJxsJa66ju+uxe2uDubn51fn705F+8/j5ODWVgUFBem6WXc1NKfT2STdbFNxthHY8xHYfyfqCia4X8ArKiqIiIggNDS00f3UvZ3RaDxGr2gwGDAYjY0ivpIkYTKZKCwsJCgoqBaBdmezGyQRg0vj6u9jRZYVvL3MKIqKxXQkqcxNCD748S9enPkLgiCwKylTN4WVRBGj0cDb46/H4ZS5vG9H8orL6dcpjt2JmXSMCSM2IkSzhBJFLCYjAqCoCu88eJ1ryf0LcovL+OjRm/nhxXv1Nn458Q4mfb4Yo0Fi6sM3kJCRr1cNm/rdH8SGB/PIDYN48qOfAO15rbI7SMoqIDTIh7KqGn5atYPWESG6z+3i1+6ndWQIAA9fNxAfF6Gsa6XSE05ZqVUwoS6IgkBYoN8x+xAFoZbrgxuyokVgPSP4qqpSbXew/XA6XmYTDqdMVkEJfdpH68R678xJPHbjpQBMnL6Qd+b/odv9BfhY2T3zWe4cfgGSJFJWWc2lPdrSNqqFfhxFUdlyIBVZUXWtsz7WAWk5RciywqG0XG5/dRZlldXYnTLX/Hc6f247hCgKPPXJAmYsWYskiuSXlHNxl3gURSHQ18rq9/6PiaOHIcsKG/dpetg2ESGs3pnAm3N+06+FyWSqcw4xmUzH5F5UVVU1eH/c1+5M5GD802g2s0x9Gtijl+9rampITEykqqpKdxo4enA9Nlv19EkAGoOjCXB5ebnu8Xqi1cEag3+zBvZ0PtwGgwF/f38MBgM2m42wsDC8vb1r6WYzMjJOiW72TBHYxppnn4/AnttoSAMLR+5/ZWUlCQkJOBwOYmKaXp67oeO5CaDQyMitKIqEhYURFhamf28wGJBEkbW7EwDo3iaKqho7Ab5W8kvKCfKzUVhaToCPN4Wl5fRyLVED3HhJd6YvWUNOYSl3Xn6BKzLrrsylYJBEJFFw7bOGsCBfjAYJP5tWZcxd1hXAZDSw64tnuLBTLL9t3s+f2w9x0/NfICsK4cG+/PDXdj5euIp2rVrSLS6C+Su30qd9ND3bRmnJTILI57+sY3dSJv+5fiCznr1duzaCwJv3Xcvf7z1KiL8P732/kvFT51NeVUNuURnjp85HdlXqKq2o1ttzdFTSE27SKoBOvDyjprKs6J81WcPsUfY2s6CEtNxCRFHk44V/s3TjPgRB4K25Kxj6xIeaa8TBVP7z/ncYPIpMHPjmOeY9PxZRFOky9jVumPwZJqN2n8sqq7lw/Dt8/staOkSHoqqwelcChWWVdG8dyR9bD+jyEEEQmLFkLeVVNYSHaIGjnOJyyiqr8baYmPvHFtbvTaZjbChZBSXMWr6JFgFa0tmVEz/h1ldmIooiWQUlhAf7YTJICAK8O34kl/ZojcVkZMXWA3z/13aSswtQFAWj0VhLF+6eQ9y5F+Hh4bpneXm5ptPOyMigqKgIu91+zPX8t4zDzf7sPEmYO7tUlmViYmKOKSlYH0k92QjqqSSCxcXFpKSkYDQaiYuLO25JxdMRgW3M4PNvJb5NgZsk+/j4EBYWRnx8PJGRkfj5+WG328nNza1VlvBE/GbPNg3sv2XgPI/a8HSEKSkpISkpSS/P7S4WUx8ao9Nz79/tvFJVY8fQCBcW9/ee8ivNFF+L4M1fuQWAnu2iySoowcfqRVZ+Mf42K+m5RQT6enM4PReLyURydgGvz15OWJAfL909gv2pOYiCoGfb73CVh5XlIwU/WrUMQkCgY0wYAJ8uXsPgx97HKSssXrebq575hNwiLQu/b/toOkS3xM/bgsWkRZj7dYzhrTm/kZFfzEv3XKUnOCmKyvWTZ2AyGvj2ubHc89ZsDJKEr7eF/333B23HvMjyzftoExWCLCsM692en1+7H5uXmf2p2bx6z1V6NStf7yNzjPv5dWftu+EumrA3OYvc4nJkRdHP2+7UikHoTg0et6ShSG1dtlcA4UF+RIRoLzyP3zSEp27RHA1uG9qbP6c+gtEgMenzJazdnYhTVli57SBj3/yGrPxSTEYDsqLw/iM30rPNEa21j9XC1hlPc1W/znpiWK+2rXhy2o9U2h0M690eURTZl5INwK1DelFVY0cSRWocTgZ2jefHl8chiiKb96cSGxZIsJ+NH1ftQFVVLuocR3J2IcnZhdzqSka78YUv6Tf+XUSXHdjIAV35dtKdAPy2eT+yorA7KVP/vr7kRlEUsdlsume5u+KmLMvk5eWRnJxMcnIy+fn5VFVV6dFX92/PZTT7s3Nnv7qdBiRJqjdqebTZsRunQsMKJ07q3BHgvLw80tPTsVqtxMXFHTdT91Qc+7wG9vTiaLmK1WqlRYsWtXSz7gxtt+YpLy+v0brZM/Ui0VQJwblcvvDfjoYcYQoLC0lLS8NisRAfH1/vC3hd/dYdwSsur6qX+LjH7fwSLeGlsW4GkiTp23r+5ud1u6m2O2jXqiXpeUVYLSYy84vxMptIyy3Ey2ykZaAfkiTyytfL+N93K0nIyOOu4f24uFtrVFVl494k9iRlEh8ewk8u3ers3zYBsNHlCaqqKrKiEBHsx5v3j8QgiXyycDUb96cS4ONFVkEJZpOBFe/8hwWv3ktVjZ2vlq6nVctAZjw5mic+/gmLychNg3q6roPAFX07MX3xarrEhfPK3Vfr12fciP4E+lixO2UkUeRAWi79OsbSs20rAAZ2a0OAzxEjfvmoeVFVtcz+rIISEjK0UuvJ2YXkFJXRMSaM0EAf135z3DdTs7BS1Vr+rLLsSppS1GPup3s7WVYod1UrS84uoPf9b+FwyhSWVnDn619T4LLY2rAvhZjQIIJ8tXl94av3sfqDxzBIIq9+s4zfNh+gRYCNlJxCVmw+wKU92vLMbZcDEDNqMqNe+AIvs4kQfxtZBSWMnDQdpyzzxYQxWE1GvZ/kFpcxbspsfKwWgvxsZOYXE3PL8zzx8Y96299/5Eb+nvp/iILAgtW7CPK1EuhjZenGvQAM6BpPTmEZh9LzuGu4Vm73hsmfM27KHL3fPzRyID5WM1dd2EXf76H0XEwmc4NjrCAcKW0cExNDTEwMISEhSJKkP3vbt29nwoQJfP/99yQkJJzTwaZmT2BBW65KT0/Hy8uLuLi4Rnm8HU1gT1ZCACdHJpxOJzk5Ofj6+uoer03BeSLaNJwpj9b6juGpm42OjiY2NpYWLVpgNBopLi7WNU/Z2dkN6mbPNg3sv+XN/zxqw5PA+vn5ERsb22RvV3dELiOvGKhNNI/WZZZVVtX5eX1wJ2255wZVVbXAR3kViZn5hAf7k5VfglGSyCoowWSQyMgvxmQ00KqlVmjBTczcekdFUdidlEW31lHM+X0TNi8zuxIzkCSRIF8rOxMy6NW2FQhgd8jsSszkqgs7M+LCThSUVrDlUDr3X30RXmYTr3/7G5c/9RHF5VVIkkRiVgFPfbKAhat3MrRXO+Y8d5d+Ls9/uYRth9K4+8p+LFy9k4qqGob0aofd4eSxD3/AbDSw+sMnuKZ/F/7afohBj75HUlYBgigwa9mGY66NZ7a/qqqoQF5xOWFBfsSFaxXEKqvtfLl0fa3IrMVoZO4fm3WSKgoCiVka4a1xJb259kpecTnVdod+DEkSmb54Dbkl5XhZtASwmNAgNn+uAvD2AAAgAElEQVQ6QffhtVqM+NmsrNx2kJGTprMjIQOAF2b+ws6EDEyuZKpf3niQrdMn4GU28cKXv3D7a7PYdihdP/6GaU/SPrql1kZRJDTQl/9cdwlPf7rA1bngg5/+Yt7KrVzSrQ0f/t/N+nUJD/Zn5dRHeOT6QciKylUTP+GdeX9gcCWm/fz6/ax4+2FEUWTZpv1YTAZa+PuwYusBAEb060RVjZ0N+5I5lJGna3fHXtGPfTMn1boPbSJbAOpxX8o87QxNJhMBAQFERUURHx9PaGgo5eXlrFq1ih9//JGnnnqKgQMH8u677x4zh8yYMYOlS5eydOlSZsyY0eAxT2T7M4FmM8vUNUApioKiKNTU1ODv769nlzYGRzsR/FMRWE/rjqOTDJpy7BPBv1kDe7bBaDTi7+9PZGQkcXFxum62vLy8lm62Pq/AM6WBbayE4N/8cvRvgyzL5ORo0Ti3TdaJjGE6gc0vqaVtrGtbcyNWp/T2KQr5JeXH7MdNZhKz8rF5mcnML8ZgkMgtKsVgkCguq9T1pE5Z5qNHb6ZHm0hiw4NRVZXcojK+/W0jVouJorIqFFVlSK/2bNyXzKU92/HERz8guWy4Ln1sKpc9+SFzf9+CJIoE+XqTOHsyE0cPpdyVYFVQWoGfTato1SkmjC3Tn2bpxr0oLs/WdXuSSMzMZ9LtV/De9ytxOBUWvna/S26gnc/BtBxuev5zvQzuRV3iOTz7eWLDgkCFMZf11a/B0WN/Sk4BgiDw87rdjJg4Dacsk5xdyOK1u+kSF87Ttw5DEkWufPoj3p3/B3Hhwdx8aS8kSSTFVTTAZDSQVVCCwSBp7g45hRSXV9EywAdJrP2ycfeV/QgN8MEpK4ycNJ3v/tyGoqhU1TjoEB3KtMduQQDaRrbgpbEj6BIbxvw/t/HxwlXafQEmTl/E5oNpBLmKLbx49wgeu+lSusaHs2Tdbp7/8meC/W28cNeVmmzgvjdZsHonQ3q148NHR+klccdfezG7kzLZn5qDyWBAVVW63/M6nyxaTaeYMKJaBrBk7S427k8hu7AESRT5+reNrN2TRHiwppOd+fRtLHr1PgySyIotGoGNDw/mrx0J2J0yT44aDMCgR9/nrx2HdALu2U81OUHDwav63AUkScLX15eLL76Yb775hltvvZWOHTuSn5/PzJkzqays1LedMmUKvr6+DB8+nOHDhxMVFcWUKVPqPWZTtz9TaDYE9mi4nQYURcFkMhEREXHCxO+fkhC4LWYcDoeeaNDUif9Ua2DPE49ThxON8kqSVKdutqamhpycnFq6WYdDi2qcj8Cexz8BT6s/AH9//xPui+6+45nkVB/c7gDu39SVyOKGIAhc+99PScjMp6isgrTcIv13Y4b1JTWnEJPRQHZhKaIg6MQi0FfT7todTkZOmoHDKTN38ljudlXLCvH34bfN+wC4fmB31u1OpFe7Vkxb8Ddmo4FFrz+okQ1B4LV7rwVgcM+2FJRW8PB735FXUo6XyYSXycjPrz/AsrcewmQwcOXT03jl66VEtQjg48dGuUqqCvRpH82iNbtwygofPjqK4vJKl55X5P8++I6qGjsLXr2fec/fjSiKTFu4ipScQry9tAinKNa2wBJc+l03Il2606v7d2HDtCcxSBIL1+zkvrfnUFZZrf/2p1fup2Wgj66LVRSVP7cf5K05vxHdMtBFVkWqahxEhPizctshl52WAVlW+HjB39z/zhxEl/bZKEm8M/46RlzYieLySnqMe4Onpv2kEeDcIoL9bTxwzQBUYFjv9rz74HV0jY/gh7938MWv6ygs08qy/vj3DoJ9vZk4ehiiILBuTxLTFq7Wos+utn//4jhen71Mjxg7ZZlBj04lMTOfF8eOoG1kCKKoteuPqY8wa9l6/TyvGdCVzyaM5vGbB1NaWc3kL37moanzUVSVrYfStEpmrSNRVZX3Hr6B5VPGYzIa9D5yafc27EnOYm9KNpd0a1Orj3pGR40uy7P60Jh5JSoqiquvvpopU6awZs0ali1bVkuLPn/+fK644gr97/79+zN//vx699fU7c8UmuUs4+k0YDKZMBqNjR406yJ6/4SEwOFw6Ofg7e19woP+P6WBbc6R2zO57H6yx/DUzcbGxh6jm83P15bsioqKdAH/6cB5Ansebrj7gKfTQHi4ZoB/MuOou+9k5BfXe0w33CVYZUWhsrKyQTcPRVE4nJHP2t2J+Fi9WL5pr57ocvVFXUnNKUQSRbxc9lLuylcDusYjywrLNu1n4/5UPl64igAfK0aDxOuzl2I0SFw7oBvbD6XTt0MMP/69HYvJiMVkIDO/GLNRYvWuRH74azuXdGvN5xNuI8TfxsylG5j/5zZyi8oAlRVbD9A1PoKwYD9kRWH+82N5ZvSwWuddXWPHIIk8etOlCAJ4mYyab6vr+4mjL+OuN7/GaJCwmI3IisLIAV2JDw8+xrdVVVWKyipRFJUau5Mbn/+M0grNHurWl2eyyuWrqigqj954KWnfvYyP1cLE6Yu45tlPqa5xcOuQ3no0dsO+JMZecSETbtXaLIoidqeT/362mJ0J6Vw/sBsA6/Ykcjgjj/EjB/LJ47cgipp+9evlG4kPD8FqNuFvs/LkLUN48pahVNXYGfHMNG556UsUReVAWi6iIDDmsr4IwNBe7Zh8x3Au792B1bsSeHDqPF6bvdxFzFVeHXc1yXNfpHVECHN+38zbc38numUA6z9+ElEUWLM7EUEQWPL6gyxet0tv++zfNvHwe/Px8/Zi1QePI4ki/3lvPn9uO8Q1/bsQFuTHLxv2UFFt59kxlyGJIpO/+JmLHnkPu1PmcEY+6XnFdG8dAcBbD4zk8DfPYfMy4+/txQ8v3u3RN7V+OPePLbrNFjSs7W7MvOLpGhMYGEhYWJj+XVpaGqWlpfj6+uqf+fr6Ulpayp49e47ZV1O3P5NodrPM0U4DRqOxSYNmXdueKglBY9tRXV2tD/zR0dFYLJZ/hBCej7Y2L9Slm3W/VZeUlJCWlkZiYiLZ2dmUl5f/I36zjbXbOo/mDU+ngfj4eL0fNmYcO/oF+OiX8Iz8YyOwR/cnH6tFj4y5y8h6wnP/BknimduGsX5vEgZJRFW15XZBEGjXqiWpruXvFq5KWu6s/PbRYUiSyE8u4/v5f24D0OQEP/1NZbWdUYN78/P6XfhYLSRnF+CUZZ4fexUtAnwBgV837OHRD79n26F0RvTrjKKozFy6AV+rhR5tIlm6aT93vjGbimo7Tqd2PjbrkXKp7gipxaNQgMVkRJK0zPVqu4O/dxwiLMiPH168FwC704kkioQF+dV57UAr2KCi4mU28uXTt+HrbSEhM5/9KVn0bhdNWm4Rnce+yhe/rNMtrv5z/UAqa+zI6pFErS8n3s5DU2tH4q58ehol5dW8O/56usZF6sfvHBvOYx/+gMMp632gX4cY/t5xGEVR2XowjbS8IsaN6E+wnzfeFjOdYsN56/5rkRWF0S/P5Jr/foqsKBzKyEMQ4OHrLkEUBTq0CuWWwb2YcOtQ8orL6Hnvm0xbuAqzUZMDdI2PYPbvmziYnqe/XKdkF3LHa7OweZl59IbBiIJATmEptw7tTZ/20Ww9mKZrnUsrq7n15S8pLK1AEARuvqQHC16+l2v6dyExM5+N+1O5fVhvTAaJqd+vZNBj7+NwyuxJymL2b5v1l6KIEH8u7tpav1buaO/ooX24sFNso5Jg3T63DaGhQEJaWlotMuqJkpJjn72mbn8m0WwIrCAIdToNNJV81jWpnwofV2jc4O0m4IBOQE7m+KfDRus8Tg1Od6KY0WjEy0sbGGNiYggLC8NqtVJeXk5mZuZxdbNNQVN8YBtji3QezROqqpKbm0taWhpeXl7Ex8djNptr2Wg1BFEUMZlMdU7Q7iHMMwLrOa55JudKokh1jR1RFPH29tZzH474sdYeDy/r3YHdiZkABPp66W0VELC4yse28NfsidzVuERXctJel7XS1Rd2BjRCHBsaxNKNe4kNC2LF5v0A3H5ZX37fcoAQPxuyomB3Only1BCq7U4eem8+Klry0uLX7uOriWMwSJKmq/z4MbwtJgwuvaz7WuQUlpKYma87GHheD/f5WUxGBnZrU+s6mTzyQDzvh6IoFJVVUurK+j+Unsem/SlYXde1c0wY2z57BovJQFpuEaUVNQzt1Q5FUfnop78J9rOxfMpDBPp48/SnC5g4fSHBfjY2fTqh1rWeOXEM46fO0y2/diZkcPeb3yCJIkveeACjQeLvHYdYunEvQ3q1Y8aE0YiiwJhXv2LAw/9DVhSqahzIssJ3z99N6wit2ILRIDL1oRsQgLve+IZLH3sfWVHIKSzFZjXz/n9uxGYxAQL+3l50bxOJJIlkFpTQITqULZ8+TftWLVm/N4nVOxMYPbQ3nz5xq24HJssKb839jTW7Erjz8gvo0z6a4vIqisur+GLCGBa9dj+Bvt5MX7yGBat30b9zHN4WE3ankxA/G/de1Z+qGge/bNjLmGG9MRokpv+8lne++wNZVvhx1Q5yikprXSt3Ho8bVXanfv/rQ1MisHUR2NLSUkpLS4/53B1VPdntzySaDYHNz8/XLabcgyYcsdFqLDwfdM+yg2dCQlBSUkJycjIGg4G4uDidfJwK/BulACeKMyEhONP2VgaDoZZuNiIiol7dbEN6weMdpzFv/uflA+c2ysrK9PLcbuLY1Bfpo589VQVcnxWUVOBwytrn9WwPUOPQJnvP/uZOBFNRUVSVvcnZ/LH1ILFhQSRk5iMrCmGBfpRX1VBYVokoCjxwtVa3vkWAD7KsEO8qC6u4qkP9NfUReraJ5M7hF+iZ9K/eew2L1+7EaJC4qEtrEjLyGNyzHdEtA0GAb1ds4pWvlxLo682To4YwaczlSKLId39uIzzIj34dYwD4+LFRuuepO2t+6ca9rN+bTMtAX+IjQvTlfEVRXLIDzyvjdlQ49ro7ZQXPT0VR5LuV2zAIAqWVVVw/eQY5xeVIksjdb37D3W9+gywrqCr07xxH0twXaNUykGkLV/PiV78ycfoi/VqPHzmQJWt3seVAqu5CcN/bc5jwyU+0CPBh3uSxroQkga7xEdw6uBff/r5Z/73RYOCeKbPJKijRo7mrP3iMn14ZhySK/N8H39Nt3Bv6vTcaJDZ9MoGebbUkZ7vDydSHb0ASRca+NZu+D05BlhWqHU5C/G38/cGj9OsQQ2llNX0emMILM3/R/Wm7xUewYPUOkrIK9BK4b8xeTnZRKW89cJ1u1QWQmlPI4MffJyW3iL4dtHv21pwVPPDuXHKLylCB9q1C2fH5RNpGtqC8qgaDJPLQyIFU1ThYtGYX7zwwEkkS6dk6kpYBRyKZFRUV+r+amhoURcXb5chgNNafjN4YAtvQHFdfNPVomcCJbn8m0WxmGl9fX33p1PPtvakkrK6lqzPhQlBQUKBHLeLi4motebnPoblYYf1bie+J4J9IrhIEAW9vb10326pVKwIDA3XdrNv4Oi8vr9G62caQfncfPh99PXchCAKxsbHHOA00NgJb3z5VVN3/3ikr5BW7iFodfVM325fr9vV27RVVVVm3N4lVuxLwMpuIDw8iPbeI6NBAdiVmYDYaKCqrpG/HWKpqHIQH+5NXUkZseIieODXn9y04nDKLXr2PVi0C+GbFJv7afoh+HWNZtTOBqhoHtw3rw6I1O7FZLbSPDkUSRYb17sAXv6wjKauAR2+6lMv6dGDjvmQeem8+7//4F5IkkpCZ52oprmunEdHD6Xm89s0yXSIhywqPfvg9e5KzCQnQosSqx/lrjgpHnjn39UnKymf7oXS96ADAfddchLfVgs3LwrK3HuLqCztzIDWHJev3cOsQzVHguskzuP21We73CR64dgDvPnQ9z995BRVVNXz563qiQvzZ8fmz9G4fTW5RGfml5Ux7bBQh/j66XZWsKAx+/D0WrN7JsD4dGDeiP+l5Rfy6YQ8Xdoolbd4rmvTh7+3897PFBPp606NNFE6njNEgcaGL5E/9fiVDn/hAP1eALdOfZqBrKT400Jcp92sk8c7Xv+bSx97T7b58rRZS576kuRAIAi9/9SulFdW8Pf56YsOCEEUBh1PmqVuGsnpnApIo0j46FICFa3bSNT6Cxa89gM1i0on21hkTeOGuK2kR4MP/5q/kyqen6TKLEH8bh76ZTGxoEBaTgT1fPsulPdoCEB0aqN+jysrKY8ouQ+2+Xl8goCkR2LpWOuorS1/fd03d/kyi2RBYk8lEWFjYMTe1qRFYT7hv7ul0IVBVlezsbLKysvDx8akVtTj696f62I357XkienpwtvizCoKAxWIhODhY182GhIRgMBgoKipqtG62sQQWzidwneuQ6qiA1dhxqKHv3btUFJXswjJtW+FYn9fCUq2IQV3FDtybiYKAJIpsP5zO6l2aZOu2YRewNyWbAB8ruxIysVpMBPhYURQFL7ORuPAQWgb4YpAk0nOLAHj566Xc8frXGCQtyigJAss378NkNDB+5ECWbtxDm8gQWkeEIAoCSVkFfLpoFeFBfsSHBfPoh99jlEQMksjHC1cBcNfwC3A4nMSHh6CoKrN/34KsKJRUVJGSU8jD11/C4tfvp6rGwdg3vqaovJKpD9/I8k37UF0kVxJFqh1OnE6ZpRv21uL57mvVOiKEnm2jkESRkZNmcMWEj5EVhR0JGSxZt5uoFlrkt21UCw5+/RxDerVn2+E01u1JonvrCAySxPTFa9idmMmYYX3w9bawcM1Onv50If957zvtWMDSDXsY+vgHJOcU8NQtQxFFgbl/aMT/1zfGs+Nwmj6u+Fq9mDh9Ia99s0y/h4qiMmPJWlJyClFVqLI7+OTxW/j0iVsAmLFkjatfKMxbuZXLJ3xU6zy/mng7Q3u1A2DUpT2Z+pAWmX3w3Xnc+/YcRElEdM1140dejNFwhNS5r4ckiUeKRLj22zayBe989ztRLQII9vNmzu+befDduRgkifEjtaj9vJVbOZiWi93pZOP+FEa/MlPbhyiwenciVotJr1Dm2YdlWT6m71ZUVNRbbMkTnj6w9aGh8bpTp07HLP+7o6mdOnU66e3PJJrNTNPQBH2iBNbTBPt0SAgURSEjI4P8/HwCAwNp1apVg96GJxOBPVVEtLy8nLS0NHJzc6moqDgnCe65JiFoynkYjUYCAgKIjIzUja/r0816DrKN0cCeJ7D/bpxMMEFA0PuNrCjklZQju4iaGwWllcf8xi01cMO9vTsiqS3VqzicMj3bRnEwLRez0Uiwv7e+rftRFUUBFfh00Wp8vb3YkZBBp5hQfnxpnB7lHHvFhbw67hqqaxzcefkFLF67C4MkcfVFXQEI8rXyxS/rAJX//ecGNu5Pxu5KWnp3/PV8+tgtBP8/e+cdHUXZRvHfzGxJ7yQkIQ2kgzTpIFWkiIo0FUFREBREQRD4FKzYUBQQUEABC02p0qQpXem9kx5SSCGbutmdme+P2R0SmqGIgNxzPLLZKTuz78zcfd773OvtoU8RT1q0iSyH3+y7c1bTZNAEDpxO0kIXjAYysvN47M1vMBokhnZv7Wje0uyf3p29ivN5BbR9oEqJc7DvVAJthk0qoUX/8c0+zH+7r07sxv2wBllW+GP/KSb8vBEfDzdEQaBW+VD+mjqcQY8/SGxKBu/MXsXzn/6EXZYRBYGn2jzA718O4d2+nZAkEbus0Kd9I2aOeBoP1wsaZQ9XM13HzMBsMjKmT0dEUWTq0s24uZj4a9oIhnZvhSRpTWjdW9bh7KJxRAT5MXXpZmr3+5jDMcn6eNg/czQrP3oJURT54ueN2OwysqKwaNN+avf7WJM9oIUvdGtRh5rlQ5EVhXH9HmHykG6IgsCK7YeZunQLfl7u+Hm5I8uK7j7RoEqEdnyOMVNYZGP3iXiqRpRlWLfWjrEh0r1lHWpEBXMqMU0/zh1Th7F7+ghMBgOfzF1H9Nl0ZFlh5Z9HWLH9sL6crCilsnwrfg1drQJb2iauKzWD9ejRg9WrV+uvV69eTY8ePfTXCQkJLFiwoNTL/1u44580zun/65URFN/G9eJyJFKWZeLj4zl//jyBgYFX9Xj9txqxLt6vxWIhPj4eu92OxWIhKSlJT4O62V3tV8PdRJpvZ6sup/F1cd2sl5eXrps9c+YMCQkJZGZmYrfb/7Y5616IwX8b1zujo40rdPN9WVHIyM7j4inVHUc1myE/L3esRTbMJiMpGZdabhVHr7b1WTt+ECoqUWX9OJGQiigKPPFgHRRFZdexWARBYMuB0wyd/AuiIPBI4xp4upnZcvAMC8Y+j11WiEnOKLFds8mAv7cHX72mPcRtdpn35qzCy92VsEBflm07RJ2KYXw97ClcTEYWbdqP0SDxeHON6MqyVnkc8sSDDO7yIACHopMI9PWgelRZrDY7BdYifnmvH3XuC9MbjaxFNpoOnoAlr5D3n3+EX/7Yh4B27S34fQ/zNuymTsUw1n32CgA/rd/FmG9X4O3uipebC7Ks8GH/zswa9QyCKPD2rJV8vXwrNrvMruNx/LBuF1HB/riYjJT18+L9Fx5hzujeiKJIl7dmMHPlDqpHBuPrSBqLevJtfly3iwZVIwn08WT6r1tZ+edhHmlcg69e66FFxoqC/p12f3smLiYjZqOBwzFnKf/UO3z801r9GFrXqUS5AB/uCw0gNiWDZz/6wZGIphGxrZOHsnb8ICRR5ExyOu0eqIwoCsxcuYM6/T+hyGYnt6CQH9ftws/LHReHNdojTWrw0mPN9e9PkkQ9YU1RVT0hS1VVXExGHqgcro9NWVFYsuUAR2KTGdSlBfdXCGXPiXg6vDGVuJQsvNw014r5Y/uy/athSJJI+wZV+ah/Z308OwMxgKs21Nrt9gs64StYaV2rjdblMGLECCwWC2vWrGHBggXEx8czYsQI/f3t27eXSNv6u+X/LVxb1t9tiOIkrLQPTlmWS0zj32wJgc1mIy4ujsLCQkJDQ/H19b2m9W9k39eLrKwskpKScHV11T0dCwoKyM3NJTc3F4vFomsrZVm+ZWT2TsWd5DULF3Sz7u5aoo3VatW/e6fXLGjNlO7u7ri4uFyy39Kmdd3D3YnrrcBeGMPOngSBDEseolByHH2/fg+PNKqKIGhhA7KikpKRTblAP21aXbqwvJM0yLLC3lMJ1K8SgY+nGwVWrQnLWmTHbDJgVxQORSdx/32hPDFmOsEB3rqfae929RFFgf9NX853q/9ky6TXqBwWRH5hERmWPEL8vfUOfkkU+OtYLHZZYeyzHRk4YR6PNK5B58Y1OBKTzMtfLiR+wXsoqiZDkCSR79fupFmN8pQPCQBg7WeDycm3YpAkHh4xGVVVWfpBfyYO6YZdVpi2bAuDu7RgxvCneGbcHFZ98jIDHm2mX4dPNK/NUx/Mxs3FxKNNaqIoKrkFRXy7egejerUjr8DK8fhUWtauqOtrm99fgQZVIjAaJN7/YQ07j8fTs1VdMix5WG12XujYGEVRSc20cNLhw/pCx8ZIosj9FULZPOk1vTIuCAJ9Hm5I349/xM/TjQZVIxEFgUFfLmBwlxa82ac9OQ4HBFEUqREVwpZJr+FiMmia4KR0qkUFs2nSawDMXb+b1X8d5fOXupBXWMToGct5tWsLKoRorgRvPNkWxaHv7f1QfZ5r3xCT0YDVZtf1saA1YoUH+ZXQCReHdJn7lbOBT5YVTRfcvFYJUlinYhjfvP4kwf7e5FttfDx3HaOeeggPNzPJGdm6hZlz+8Wtry4nH3DCarXq3ORGNLB/V4EF6N+//xXf69mzJz179iz18v8W7vgnzbVYWDlxcSPXjTRRObfh3K4zZMFqtRIREfG35PXi9W9k39e7bnp6OklJSbi7u+tNcqIo4uHhQdmyZUtU5woKCvTo28tNNd8JuJsqhf9E01Rx3WxkZCRRUVG4uGhVhszMTF03m5qaWqIyf88D9r+BG5FzleY+ZZAkMi15lxCOg9HJJRqXDJJIamb2JeQVNNKwaPMBJEnU048Apgztiawo7D8Vz+nENMoHB7Bx73G83FwI8fdm/Lx1pGRqWj9PV23MO6uSFR12TiajxMivl+r1YeeU7uAuLVi8eR/Vo4IpLLJx5qzmOSoKkLrkQ8wmA0ditGPYeSyO4dOW4u5qJtOSR7NXvmTd7hN4upmx2bXp7MMxyexxeJGCJjeYsnQz1SKDmTDoCf08xCSn02zwBHLyC1kwpi+PNKqBIAjEpmQwoHNTTv8wFncXE299u4Jub3/L6aRzCIJATn4hH7zQmUea1AS0qvCY3g/jYjLy/vdraDxoApmWPFRUgv29OfDdaBa801dLBRvwKV/8/DvlQwKoW6kcuQVWurw1nQKrjZ/GPEfDqlF61fGTAY8xfNoSREdTFUD15z5g8uI/qBQWRLkyvsQkp9N40OeM+maZPoZG92pH7Px3CfDx4Oc/9rLg972U8fEkO6+A5kO+4Hh8CqKoxbaqqBgNEnkFVjzdXAgPuvDcdVZarwXOseccV87GNFVVSTufQ0FREaFlfDAaJFxNRt7t2xEPNzOKopYgr87z6hwrzuTEK6E018/NqMDeLbhjCOyVvojr6X692Errekjw5T5bYWEh0dHRKIpCVFQUnp6e17T+rZ46d+4vNTUVLy+vSxwein8+d3d3goKCdP9dQRAumWrOysr62wv0v4I7rQJ7NRiNRv0/p27W1dUVi8Wi62ZPnz7NnDlzOHLkyFXH8fbt2xkyZEip9jtjxgzWrFnDmjVrSkxn3cPtieuVYjkLCM5RbJQk0h2NWsUR6ONRgqiKApzLsqBywSdVVhSd5I6dtZJTSefo0rwWCzbuQRJF3MwmJFGkbIAPB88k4ePhxoY9xxEEgZe7tADA19PNcV1p+zk+5y2+G/E0oigyYMI8RFFkWM82TP91K6qqkl9YhKKodGhYHS83FyRRZO1nr1A5LAiAqpEXUpDcXcxIksjnCzdS5/6CgRcAACAASURBVL5yBPp4MGftTk4mpukBBLKisOrjlxjes7VeSTQaJH7/cgjjfvyNnPxCKpXTKsHzNuwmKjiAIV1b0mvcHN3S4K2Zv9J08AS+XbkNs9mILCs0v78ClcLKEBXsz9KtB6nR90NOJqbpTUsrP35Jn2aPTk6nT7sG+Hm5M37+BjqMnIpBEjFIErKi8O0bvSjr64mqajplD1czE1/pxuYDpx1jQWtoe/GzuUiiyK8fDdS/a4BdX7+Bl5uLTgyjggM4NudNxvXrTG5hER3emMq3q3ZgdmiF+3ZoTPIv4/B2d0USRcb0bk+VcM0tIDEtC3cXM6IgcNYRQ+wk/Rf75/4dFEUt4W6hjzXxQr9MoI+n9j069iFJ4iXa6xL75oKW+0afj9diZ1ia5e503PFHd6Pkr7iu70YJbHp6uh6y4Obmds3buZUVWFVVycnRrGq8vb0JCwsr1WAXBC35RhTFEhZNsixz7tw5YmJibshvtPh+/incih8KdxOBLb4fp242JCSkRGV+69atzJgxg3HjxtGrVy9GjBjB7t279fWPHDnCjBkzmD9/vj7urobx48fj5eVF+/btad++PWFhYYwfP/6fPMR7uEHcUBOXIOhVKpNRcmhgS6Jjg5LNSpIkkZaVgySKHDyVAKBbRtllmXPZeSzffhgvtwtJh4qismHPMQK83TkaexazyUDSufNkWvLo2Kg6ze+vgMkg6aR6zHcrAagSUZadx2NZvPkAv2zaR52K5Vj911FyCqy4uZh0nWf7htWRZYUAbw+9Oay4W0Kkw0rpp7f6sPQDrTnsp3WaP2q7B6qw91QC7UZM4dz5XIb3bIOqKjzw4ie8N3sVoQE+bP9qmL4/g0Hk03nr2bj3BN1aaslQkigiSSJv9+3IE81r0aJ2JVDhcEwyvds14I8vX0UQBD5bsAGbIhMe6MuR2GQeGDCe6LPpOpldO34wnwx4DICUTAuBPh4oisrGvSf45Y/93F8hlB6t6yIIAkMm/cz4+esJD/LjkSY1EAVBD3n4whE84Nzu7/tO8oxD5tD74YaIokDDl8bzzqxV+Ht7YDRIeLqa6dCwGg/efx+CKDDoy4V8/9tOECCvsAhPNxfa1rswFl7o1ET/t1OOoY8Rh4yktBVLbVxd+hws/gPpaihOfiVRZNT05RyOSS72/t/PVl7NiaC0z5XSSAjuBtxRBPZyX9qNVmCLW8JcL7Fx2ks4AwqKJ8aUBrdaQqAoComJiRQUFABctcHsSvt0EpriU82RkZEEBARc4jeanp6O1Wq97Zqz7obplX/T7cCZghQUFESvXr0YOnQotWrVwmq1snz5cl566SV92erVq9O/f3+aNm1aqv0tXLiQDh066K+bNGnCwoULr7LGPfzbuPFeAu3/Bkki4zIV2AdrRpV4ragqOXkFCIJAoePHclZOPh/9uAaDJNGpYTXW7zmBIAi89/wjjrVUfDzdcDEZOe6QFbSrX41Vfx4mxN+bsc921Bt3/jwWxze/bmfasq1IoqhXVMfPX48AjOvXGS9H7KvT2kpVVeIdFlyCoBGerJx8Gr/8OTNWbEOSRKYt28LizQf0xKyNX7zC3Df74GIy8u3KHRyPT2Xykk367GC3lnWYumwLmw6cIiLID0kUGfD5PCRBZPWnL/Pi5/MRBc3s/7edx2g4cDypmTlMerU794WWYdryLbQbMYXPF2xAFASNYH7yMqs+egkXk5EJP/9OfFoWZf28iE3J5MFXJxKflqmTzi8Hd2X2qN5aE5nNzumz5zRtqChitdmZNKQ7r/fQuvWdlcYm1aOIT83ExWTEaDRgl2VW/nmYVnUqMWvkM5rswrH9TRNf47kOjQDNSeDc+VyGdG1JhdAAREGgc+MadG9ZB0kU6Tf+J3Yej0MUBZZtP4jF8f3D5TlAaT2uRVHkaGwK+xw/hIpvy6mnBnQ7ruJEtSSfEDnjcCmw5Bey81gc1R0V+NJWX4sXfi6+514rgb1Xgb3NcSMVSCeu14RbVbVYxZQU7Ubo6+t7icdraXAriZSiKMTHx5OdnV1qiUNpYTKZ8PPzK+E3KkkSmZmZxMXFXbN5/p2Mu7UCeyUYjUYeffRRRo4cyc8//8zkyZOZPHnyde0rISHhkpQXpw/hkSNHrmub93DzcDU51w3ZETr+bzJIutdrcYQGlNQWSqJIgMPYPyJYq7wVWm3MXbcTuyzzatcW7DudSFaO1mS0dMt+BEGgXqUIJFHE6LhXP1i7Iiu2H0KSRGrdV07f9oY9JwBoXVczovd2d+XLQV2JTclk6daDVI+6IA3Iysln3A9rEASBrQdP89FPa/X3DpxO4szZdLq2qE1KpoX3vv9N07ZKIr9s2kduvpXWdSuhqCoFDvLyeo/WZOcWcCQ2hTeefIgxvR+mTsUw/dzLisLbs1ZS1s+LA9+OArTz36pORXw8Xen5zrf6/utWDEeSBDo1roGiwsdz15FbYKVm+RBkWWFEzzYsfb8/bi4mPl+4gbPp2QT5enEsLoUaz39IpiUPQRDItOTRoWF1Rj7VFhU4n1tArMOdwfkMtdrsFDiqpGGBvkiSiLXIhkGSaFGrIgAGg4SqwtHYZAZ/uRAXk5GIIF/tR8PRGCYv2oSsKCSmnScp/TztG1bD1RH3O29MXx6opDkEPNbkfrzcL6RZiqKop4LpY8rRMOfEecdYiE5OZ9AXC/TAA5tdplpkWepULFfieDIseRyJ1Z7vznugqmqaa0VRUIrdF3PyC7HZZSqUCwTAy82FjV+8gtEgac1mpewVKU50r7dR9r/SUHvHH931ks/iA8NJOq+1Eezs2bOkpaXpaRTXSyRuVQVWlmViY2PJzc0lJCQEDw+PKy57owTT6TcaFhZG+fLlCQwMxGg0ljDPT01NJT8//1/T/t7p+3DiVhDY4l20V1sGwN3dnXbt2tGoUaPr2ldCQsIVIwqzs7Ova5v38M/jRiuwzulOg0Ek/TISAk/XS2e2GtfQNKLe7ppkq7DITqYlj3W7jlEjKhiTwUChI19++daDJa6VqhFlOZt+nloVyvH7vhMUWm0IgkBSumbN9fs+Tc9ZPbIsWw9Fs/N4HD1a1SW0jA/xaVlIosjZjGzem7MKf2930rNz2XU8jqfb1qdBlQiNPIkibepV5ticN/HzdOfbVTuQFYXnOzaioMjGiK+X0f3dWZoWVBCYOaIXu78eQYC3BxMX/UGXMTM4GJ3EK11b4eXmwlvfriA108LU13qy5eBpcvILcTWZsNnsPDT8K4rsMsvGDeCHN59FEkVSMy00qRHFie/HUCkskO9/+4sJCzfy6/ZDSKLIrhNxRAX707BaBKqq8umAxzg8azRmo3bePurXGT8vdxLSsnhnzirdcir67Dl8PFypHB5U4vswGw24uphKSPPcXLTvzcPVrJPA00nnqBYZfMGGTJY5EZ9K5yY1eadvRwQE8q1Felxw9Nl0Dp1JAjSdaZHj74qqXjKt73QPuBy8PDTCWz44gClDeyIKAn8djaV2v49JzbToY3DDnhN8PHct/l7u1IjUtLbOSvvuE1qV1pJv1Z3eZFlh1V9HGDLpZ83SbONeXvj0p2KfSbyqfdaVcKMSgnsE9jbH9RJYpZixsDPWtfRCb62KmZWVRZkyZQgNDb2m9S/GrSCwdrudmJgYCgoKCAsLw8/vQmfmP022DAYDPj4+JczznU1AiYmJ/4rXLNwa4vdfqcDC1eMLrwUWi6VE6osTF6fB3MPthRutwDphlCTyCot0kuKE2XTp7FZlRyOPM13JKSXYdSwWgyTx6YuPEuyv/Rj6bnQfLA4bJ8WhV915NJbgAG9Sln6Ki9lI9Nl0bDYZS34hR+JSqBoehKvZxO/7TjLqm+VIosCS9/oz6HHNu7WMtwfzN+zhfG4+o595mJcmzMdqs9OyTkW2H44mtNubfPDDGnwciV/1KoXh4+5KhZAAVv91lLzCIt54sjWiKNB17Lf8tH43YYG+yLJCoI8nlrxC9p9K1I83NMCbp96fjayobPzyVTxczYiigCiJVAoLpOe73+FiMhIV7M+iLfup+fxHvDLpZ9xcTKBC/crhVAkLpM/DDTmVmMbjY2Yw39HgtmjzfmJTMvVzWadiOTo21JKWwgJ9+WJQV33a/z6HIwNc3/OjUlhgidcGSdK3KUkix+JTqBpRlqiy/oCmba1WrOJtcjR3iZe5J8mygiW/kEWb96GqKmfOnqPxoM+xFtlJP59Lu+FT2HroDKBN8zesFsnBb0cT5OfFgdOJ/LbzKG3qVaZv+0YO/12tse7TeesAlYrlyvDbzqP6D6r5G/ew+2Q8PVvVY8prPbDkFfDmt7/y+96T2B1EOj8/n5ycnGu+Pi6+l94jsCVxRx3d5b606yV/sizr6zqn0kszuJxEMCcnh+DgYIKCgm5aE9g/RSSLiop0a6/w8PCbUjG+3s96cRNQcHAw7u7uJZKgnDnRd7LX7H9NQuBcBm78mK9Ufb1YVnAPtxdKY6NVGjgJVGZO/mX/Xhw+Xh6oqqq/l1+oTb+aHQb2DzkiRp1wN5uw22UKCouoUSGUPSfjdLsnRVUpHxJAWKAvWZZ8wgN96N2uPgCbDpzmcGwyczfsISLID6NBYvT0ZRgNEm88/RBTlmwm0MeTxtWjiE7WfJNVRwLYT+t2oqpaA1L7BtU49v2bGCSJpVsOANCmbmWOxqaw5dAZth2KRhAE8gqLGPhYM2LmvUPvhxtwIj6VvScTGPhoM17o2JjYlAydvD045EvOpmczcXBXmtUor0+LP9q4Jj1a1SUiUNPNHo45S/XyIWyePBRXs5Gk9Gwigvx4snU9jsamMGjiL3i4mpFlhSaDvyA5IxtJEtmwV5NS6J39csnZmCvpNC9XBb2i/EQQdFIKUD0yWCd/+rqXXbOkj6tz/Pl6utGlWW0EQaBCSBl2THkdSRIJ9PXk7Wc70LhaFMfiUqj+7Dhe+2qRvvGwQF/enrWS71btIMDHAwGBGSu2I8sKAx5txshvluHr6cZDD1TRU9HaN6hGnw+/Z8eRGERRxNvDjXee68Br3VthcMgXsrKy9Ofb2bNnsVgsV5UUXKkYUNp77D0JwR2CG6nAOlFaAukkgoWFhYSHh+Pv76+vfyOk7p+swDqtvex2+xWtvf4tPaooinh6el6SBGW327VfzleINb0T8G82V/1b+7lZv/qdP7Cu9b17uDW4mgb2Zoz7cmV8AC5xIjAWDyqQS96/ndO7hUUagXU1GVEUFV8vTVpwOkkjlZIkYjBIuLu5IAoCAx57UN/Owj/28vnCDUiSSFigD5snvUrfDg3JKyzicKzWSf7zpv36FPbMlds5mZBKj5Z1mblyO9m5BUwa0p0aUSGoKjSpXp4BnZvy9nOdMEgiT38wh76f/Kifo+nDn2L5uP54uJqZt3EPAGP6PExugZUGL33G+PkbcDEbEYAiu0y/T7UGpl4P1adqRFn9uH9661meen8WBkli5NPtkESRZz/6nrTzOUx6pRvDerTidNI5OoycxhNjZujnrmXtivw1dTgGUaRqRBDRc8cSFuiLyWhg+1dDCfTRnhWtalfSz5GilvTcvfj7Ln6fKG4ppSgXfNZLM0YURcUgiSXHmqBF6Grvl3zen88tcOxTZPGWA7qP8KzVf7Lg9z0oisKZs+mcSkyjac3ySJJIpXKBTB/+FK888SCSKLJ48wE83VzYPHkoz7StjyRqBLVt3UoM+nIh3u6ujO7VTjsG4GhcMv0/m4uXuws7p42gfpVwktLPs+NINM881IBBXS6MLWfRxumlnpKSwpkzZ0hMTOT8+fOXNHg5n3dX0sDeq8BquOOP7mZWL6+2jYKCAqKjo5FlmcjIyEsqQbcjgc3PzycmJgbgstZet1MXfnGvWU9PT+2XrLd3Ca9Z58V+PVqi4riVQQZ3SwVWUUpvRXOjN83q1atfIhdwVl+rV69+Q9u+h38OpanA5ufnX/V9gIbVIgE4l5172UIDaITuYthlWddMmk0GVFRQwS4rHI1PZW+xqXjQrp1gf2/9fvDFwo3IsvbvMd+tpKBQkyMUFtkwiFol7KnWdZEVBZPRwNrPBlMpLAg3FxOje7Vjzm9/IggC587ncl+vdziTlM7bz3bk8aY1ORGfyl/HYtl66IxeYT0en0qjalGoqsqop9uybvwgQgN8WLPzKJk5+XRuUgNJFMnMyadm+RD2zhxFgyoR+uc/k3SOT+auJSzQlzWfvlwitjXQx5M2QychCDh8Vv158ZEmvNu3E4Io0Pl/3zDmuxXIikJMSganEs/hYjTqRBAu2EkVJ6KiIOiVUfmie4LdYdifkpld4r6kODx1ne4Oim5pduWxUnyfTvN+URD0yNeL7zH2YgWOMj7u+Hm5oygKs9f8SZPq5Smyyzz53neM+HoJiqKyaf8p9pyMp2Oj6kSV9ed4fAoDJ8yn06ivMRkMSAaRn9bv4pN564gKCaB7yzr8sf+Uvt1DZ5KoGl6WIF9PXp+yGG8PzZ/26fdn0WXMDGas2IaAdj04P5uHh4fupR4WFoavry82m420tDRiYmKIi4sjIyMDq9V6RQvKa/WBvZ2e8f8E7ngCe70V2IshSdIVt5Gbm0tMTAyCIBAVFYW7u/sly/xbBNaJi9fNzc0lNjYWURQpX768nqRUmnX/bTgrC4GBgSW8Zp0Xe3R09E3xmv0ncTdKCP5J8+yEhAQWLFigv+7RowerV6/WX69evZoePXpc83bv4dbh7wJhLBaL/oP6amjkILAZ2XklOsqLw2aXiU/NZNEfewGNwMiKSqEjKtbFZHSMWQFJFDhzNoO1u0+W2Iazycp5/Xi5u9L8/grIisLcjXt5eOQ0JFHE38udvdNHUDE0gG4P1tYrmPdXCNW31a9TUwY+2hxZVjAZJPIKiuj1wWzyrUWYjEZ+/fMQAMO6t0YSRfp8+AMdR3+NzS5z5mw6sSmZ1LpP2163FnVIW/IhlcppmlA/R7DCxee1QmgZRj6tVQTdXMwIgsC7s1eSkJbFJwMeZ9/MUQiCwPH4VM6dz+Xt5zpSMyoEURBoVrM8I55sgySKTFi4kdAAbyRJxCiJlzRFXSwFOHA6UavEiiIjpi0h2REe8Mf+U5yITyXI10snq79s2kf02XS9AiuJIvkFRfr5v3j7xfetqKpOlos/X+NTMwFtnKVn53I+twBfDzfHGFBo6agYi6LIpomvEeLvjdlo4OP+jzL99acpstsZPPFnHn1zBvmFRYBAlfCyHJ39Js889ACgyRJ6tKzLjiPRLN9+iLYPVNEjeFXg7dmriEnOYOyzHfQfTYIg8Fq31iiKio+Hm6ZNFkW46HsTBAFXV1fKlClDZGQkERERBAQEIAgCGRkZxMXFcebMGX354t/7tWhg7/bqK9xhBPZmamAvXsfPz++y2zh//jyxsbGYTKarEsGbof+6kQpscWRnZxMXF4fRaKR8+fJ6k1pp1i3tPm8V6S3uNRsVFUVERAT+/v4lvGaL/3K93cj4P407RUKQkJDAjBkzWLBgAdu3b2f8+PGsWbNGf3/79u0l0rZGjBiBxWJhzZo1LFiwgPj4eEaMGHGTjuYe/glcrZiQmZlJfHz8VX9IOxEaoEkI0h32TZdDnlVr8hr42U8oikpyRjaiAFbHVKyLWZMQOKuECWnnmbxsK6lZOSVIks0hVwLw8XCjXqUwJFGkXqUwujTTIlZ/3XGYAB8P/vjyVSRJ1LWagiBwKvEcj4yexpGYZIwGCUEAbw9X/vfMw0QnZ5CTb0UUBV7p0pJWtSvSpXktzp3PZduRaAY80hSjQeKTeesZPk2rDC7depA1O4+WOJ+gkTlFVUsQepvdOZ2uVSdFUeCtPh2Yt3E3oijgajahKArfrdpBrw9mI8sK+dYiZEVh5FMP4eWmdeRPHdoTd0dDkrur2WFFdamxv3Pf9SqH6/rb8S91oayfNhu571QCGZZcBEFg2+FoEtKy6NaiDuVDApAkkUWb9mOzy7i7mXX7Kacllb6vYjZYajEZQfEwgvAgP/1+tHjzAXLyCxAE6Py/bxg4YT6qqvL92p30fG+Ww5rMRkxyBg83qEaAjxaW8MmAx5jyWnfcXEy8MulnRk1fjqe7C8881IDYlAxavPoF+YVFzB/7PL9s2ofo0EgPn7qYIpvM9Nefor8jZWzq0J4UFtnYdugMjzW9nyea1+LhBlX1Y7JarVca6giCgNlsxs/Pj/DwcKKionTHHtDuqbGxsZfEdpfmXny3V18Brt209DbDjVRgZVnWLbT8/f1LJAQ5CVJqairu7u6Eh4dftbv6ZlRgbwTOfWdlZZGUlISrqyuRkZF3XRKH2WzGbDbj7++PzWYjNzeXnJwcMjIyyMjIwGg04uHhgaenJ2az+bLn9lZUR++mCqyz+nMjEoKwsDD69+9P//79L7tuz5496dmzZ4m/XWnZe/h3caVxcLligqqqpKamkp6ejqenJ2FhYaXahyhqvqOXS0UCyM4rJNgRZaqoCvEpmQT5eemWWa4mI7KicCoxjaoRwWTm5GOXFTxdzSWafsRiFVgfdxe9GaxR1QgaVYtCUVVe/WoxiWnnGfhoU6w2O5OXbGZY91ZIokiF0AB+fqcfgqAV2pw64Jcfb86fR6MJDvDmt13HaFAlgrljntOaglZuQ1FU+rZviCW/kDU7j7F2/MuIooCHi5k29bQKoqKqFNnsuDga0n5ct4tG1aIoH+yn72fh73vo0aoeoFUyJUnkjScfAjTpQ35hEZ8OfFy/fpdtPYiPhyudGtcgKycfF7MB14sKHIqjeufcnhN/7D9Fkd1Ou/pVWbf7OLtPxDHq6XYUFtmRRIERT7bVfxx8tXgTVpudn999QT/fjapFsmjzfj2UQFVVVv55hI6NqjuqlIJDOqD9+2hciu5VK4oC2w5FczoxjWfaNaDQZsdslHixc1NUVUVWVKa//hRBvp4IgsD736/m8Wa1EAWBzxduYMrSLRyb8yauJhPHE1Lp2Kg6iqpitdnZcSSaxHPZDO7SnHJlfAkP9KVepXD6fPQ9y8YNYM6o3vq5eK59Iz6Zt5Z3nutEp0bV2XEkhqY1yrNy+2FGz/yVzZNeY8prPfXzZrfL18RNjEYjPj4++PhoP+Kc0rrs7Gyys7P1sZqXl4ckSZd9vuuSi3sV2Nsfus7mOghs8XXc3d1LiMyTk5NJTU3F29ubiIiIvyWCN9LAcDM0sKBF2SYlJeHh4UFUVNTffuZ/2v3gn4bTazY8PPwSr9n4+HhiYmJIS0v7V71m7xYCC6XXXd1tP5ruoXS4uJjgTPxLT0/Xr9PSPlQrhwVeNk7Wiey8Qrw9XDE5TPHjU7XkqKJiGlhFUfnzaAwqKpaCQv3vi7YcvHA/UC+M29Z1KunXUs9WdakcFsipxHOU9fNkQOemqMB3q/9kxY4jSKLIBz/+Ro93Z6GiYjIaGD1zGcOnLdGqakYDC95+AUVReePrpbQZNpmUTAuiKNC3Q2O2fDmE8CA/XE0GVn8ykCoOP9W2D1TWK46xyRk899GPKIrK/I17GDZ1Ca4mA1abnWFTFxOXmkmPVvVQVZWN+05yNE5rNHMSRrPRgJ+XJndzHlevh+rTqXENQOvUv5i8wgVrKkkSiU/L1P+eU1BI6zqVEAWBoV8tokJIGSRR5LWvfqHiM++hKAqFRXaKbHbmje3LLw7y2vjlz/nop7WUC/SlR8u6SKLId6t2YJcV2j1QhQJrkSN1TGvEW/HnYQBqlg8BNC10Vk4+TWuWp0/7hkiSyMDP59P45c+185SSyYodh/UGNICjs9/i4xcfBWD1zmO0ql0RP093ZqzcTvs3prJ+zwlEx/e08+sR/Db+ZcqV8WXeht3Ep2UxfmAXKjoCCVRVZd/pBGav+ZOaFUKJT8niTNI5Xu3WimY1KyAIAh0aVSevoIiBE+aXIP2Kcv3Nx85GruJNzs6Uz3Pnzl2xCey/on+Fu4DAwvWTx8t1UCqKQkJCApmZmfj7+1OuXLlS3XT/bQ1sbm4uKSkpeHl5XdOD4m7B5bxmzWYz2dnZJCYmEh0dTUpKCnl5eXe0PdfFuJUE9lbZaN3D7Y+rybmcqUNxcXFkZ2cTFBRESEjINY2LKuFlLxsn64QzXev7MX0xSCIJ55zxrdo+XExGZEXlr2OaTdb9USEE+3shiSKHimXTbzpwSv/3k23q6f8OC/TF19OV3Sfi+fWDFzU9oyCwZOshujS7H4AdR2LZfPAM53MLEAWBt55pz5ItB5ixYpu+neiz50jOsJB47jw+7q7EpmSw+cBpKkc4/WsN1IgKufT5IWhT5XPHPgeCFs065dXuhJbxYeqyrfy4bhdz1v6lH3OzmhWYuWI7e07EIcuK3nTprIherGt1wqk/LS5NKP7v8MALfuGPNb1fJ4hHZr9Jz1Z1Aa1aPvKptoiiyJhvV1DrhY85nXRO8wVWVaa81oNdx+P0SipA9ahger7zLUaDhKtZI9HLtx1EFAVa16lc4ly4mAy8Pm1JidSrOaN7s91hjTVwwnxe/nIBBdYiYlMyGDplEemWXJ3Ib588jLlvPatXoD1cTDxYqwK7j8fReugk9p1K5P4KoSiKwvQV23jszW84l53LpwMecxyfSJ37wli5/TA5eYV89nIX5m7YjSSK2O0yL02Yh5uLic9efpxdx+LIzruQNHkjDcdOAitJkh7b7QwfKleu3BWbwA4fPkxubu5/ggPcFUd4s/wHFUUhNjYWi8VC2bJlCQ4OLvVN998isM51CgoK9OSr0g7cGyXOt2vl1uk1GxoaqnvNurm5kZubS1JSEklJWqJLYWHhP0Zm78YKbGklBPcqsP9NOO87NpuN6Oho8vLyCA0NpUyZMtc8RiuVK0OOI3Tgcihw2GW1faAagiCQmJapTeE79uNqNiErCruPxwHQoEoYUWU1MpaRnYfsIGkpGRemZS/+jAZJonnN8vh7a1XM87kFHIo5S+PqUciywqGYs7iYDAT5eHI0NgVPNxd+/XAAlWnqawAAIABJREFUn85fD2jXQ6WwII7MfpOnWtfD3dXM18u30efjH7Da7Ow5maAvW/xeqigqx+JSWLH9kCbByMyhjI8H3VvWQVYUHry/AmX9vBj55EOkZFroNHIa587n8MXgrtSpGO7wJ9WaseJStAqq87woF92ztUhUrdnNkl9IQaENURQY8Pl8Rn69FIBJi/+gwtPvYLPLnExI45VJP5dwIPhm2FO8/PiDFNns7D2ZgJe7C5Fl/dl2OJr5G/ZQt1KYVo2VRNoMm8SP63bRsGokv7zXH0EQ9M8W5OfFmG9X4GY2oagq+04l8MbXSxBFkVkjn0EUBN6etZJmg78gLvVC4MKS9/uzddJQXM0mPvjhN35avxtJEDmfm88TY2ewZudRBMfYXP/5YA7PehOTwcCn8zdwOCaZ8zn5SKKIrKj8Nn4QLWrdp0/BK4pCm6GTsCsKE4d049N56/DzcmdMnw6aLlcUWLf7OHtOxNOtRR2eblMPb3dXnZPciP3j5civc5w4m8CioqKIjIzUm8B27tzJ888/z8CBA3n77bf5/vvvSU1Nve7PcLvjjiKw/7T/oNVqpaCggHLlyhEQEHDNn+1WE1jnFB2Ai4vLNVc5rhd3UoWtuNds+fLlCQkJ0e3EMjMzda/ZvzOWvlbcCnJ/K0lyafbzX/EevIfLwzk+EhMTsdlsREZG4uvre13bqhQWqE9/O6uH+dYLjiNNqkWVqCrqRE3SPoOb2YjNLpOQlkWGJY+q4UFEBDr8ZS15utSgy4O1S+w3PjWzxLVbLtCXAqsNq83OtsPRKIrKfSEBHE9IpbDITufGNZAkkQk//87UpVuoFhnMgZmjtc/iqECW8fFg4pBu5BVaWfD7Xt7s9TBmowFfDxdG9Gyjn7sf1+/mzNl0QKXvJz+RbslDEkUGTVzI/c9/jF1WOJ9TQP0qEeyd/gaebi588fPv7DoRT9cxM1HRqqEr/zzC429Np7DIRligr759WVb0ZqTiP9yz8wpIOnceLzcXXF00ve3kId34yDEFv/dUImP7dMBokPjwp7Us2XLIEakaT9Vnx7HvVIIeRLDxiyFsnvQqoigwevpyXvtqEX/sO6XfEz7o15lR3yxzRFNrFeLOo6fx/W9/0bBqJO/27ehwjhCpfV85OjjS0pz43zMP071lbUL8vYhPzeKNr5cSn5pJRJAfsqLwxpNt+bj/owT6ejJt2Va2HoomxN8LVVV59qMf+Gn9HsyOKvKc0b35/n+9aV23Mut2H6fTyGkUFclMfKU7wf7eHItLAUFg2rAn+Wz+ekICfAhyNKypqsq3q7YjiSKfv/wE785ehdEgMf7lJyiy2TVHhBt8nsiyTH5+fgmnHWdlvfi92GQy6U1gzZo149FHH8XT05NDhw4xbtw4OnbsWCr7ujsRd8WT5kYiDIuv5+/vT0REhC6gvhbcagLrjLN1Crvd3Nyumcjc6RrYa4Uoinh4eOgP1YCAAN1r9mJj6Rv1mnXibmgUK62/6z0C+99G8W7r8uXL69Od14Mq4UEE+mpG+s6p4NyCCw/yIF8PvWEL4Hh8im7TBFpql5Pg7j4eR1lfTz0gITMnX/eRdTWbStz/jsWllrieREHA1WzEbDSw77RWLPB0M3M0NgWAdg9U0fZxIoFxP63jdOI5XMxG9p2KZ8S0JSV0uW5mE6d/HMugx5tr5yjkQmV62+Fo3puzmoggX1b9dZS41Cw6N6lBaqaFbYdj6PdIYwySyKtfLeKh4VN0B4I3nmxDtciyfDzgMf3YOzWqTq+29fnop7WIDuJ6OOYsGTl5yIrC2p3HLuiVVZUMSx5fL9+Koij8dSyWt2etxGQ06MvMHvkMfR5uAGjWYR/264TJaGDS4k1kWPKoEFKGhLQsmr7yBb9uP4RR0gjidyN70blxDZrdX4G9JxOYt2E3jatFEb/wfURR5MCZRPILi1j8/oscj0/VY1sVRaHxy5+xdOsBWtWpRJt6lfnraAzdxs4k6dx5Xu3WCpPRwMyV25i95i/iUjORJJF1u48TUdaPvh20CNgGVSN48P4K1CwfytKth1i985juejBt2RZOJabRvkE1APw83Zk58mk83Mz6dx/s78U3y7ZSKSwQHw9XsvMKGNTlQRRFm/VKzcph5/FY2jeszp6TCWw5eAZZVsiw5GGQpJti8XgxCf67GTdPT0/eeOMNpkyZwvjx4+nXrx89evTQtbN/hxkzZrBmzRrWrFlTwhXmcnC6ySQkJACwYMGCEs4ytwJ3xZPmRshjYeGFaSofH5/rvuneSgIryzKxsbHk5uYSEhKCJEn/GRJ6M+Hi4qJ7zV5sLB0dHU1CQgJZWVmXpKSUBneb00Fp9nOPwP53cPFYyMrKIiVFI3VBQUGlssuCK2szo4L9CQ3w1okaQE7BBYIcnZJVYvl9343BZDTQ++FGuhWT8zPuPRmPq9lIeYeEINOSX2K7zuWsNjtlfLT7f65jX8ULHPUqhdOxQVVMRgMnEtMAqF0hlHPnczmbkY3NLmNzVDnLeHsy57e/2H44GkkUmbVqu1Y5c2hAnYlhzin9xtWjODLrfxgkiXyrjbAy3pTx9mTJVq3hrOuDtckrtPLH/lOkZlkwGQ3sO5WIu6uZ9Z8PpnnNCqz68whh3cewbvdxuresw3t9OyFJop46NXzqYvafTqB9Q420rdh+iB/X7eS+0DK893wnRFFkypLNzF2/myK7zN5TCTR86TMOnE5CEASteax7K555SIvXbVA5nGcfro+XuwvTlm/VGt78vUCAuet3ExHkx4wRT2GQREZPX86rkxcxd/0uBEE7rzNXbKfTqGkUWIsY168zkiiyYOMeZEVlw4QhHI5J1r/H2hXDMBklpi3fopO4t5/ryPJxL9KmXhV+33eSPh/+wEsT5mvfcU4+bepW5pd3+yGKAgFe7lQI9qd7yzoci0vh7dmr+HT+elRVJSXTQr3KYbreV3el8HCjW6s6HI45y8DHmuPtrtmOHYtLJjXTwitPtGDSoj8wGw2807ej9kNFgGB/b2x2+z8iTyutG4woilSrVo0RI0YwcuTIUsm6xo8fj5eXF+3bt6d9+/aEhYUxfvz4Ky5vsViYOXMmbdu2pXLlysTHx9O+fftrPqYbwV3xpLneCmx2djYnT578+wVLgVtFYJ36soKCAsLCwvDz8yv1ujey35ux3u2KyxlL+/v7oygK586duyQlpTS4lRKCW7Wf0mpg7ySJyT3cGFRVJS0tTbfug2vTQF/OpF1VVQySRIOqkZzLytbft+QVkudIx/rzRCKSeOGe6+ZyoZv+o4Fd9PQngL0ntQpRzahgQPOXdZrPK4qiV0gLrDadwBZYi1AUVW8WkxWFdg9UZvaoZwB0g/0yPh4cOKNp6k0GAxXLBZCdV0C5QF+GdW/Nq5N/xq4oPNehMaBVdGVF4ZFR05i8+A9EQYs7nbJks67p7NGyDjunjUAUBVRV06lGBPmxdtcJiuyybuH11rcraPP6V7rXrI+HK1abnYGfz8dmlxFFkTU7j9Jq6EQKimzMfKOX7nYA0LRmBaYu2czppHMIgoDNLjN7VG/2fzsKk0HindmriEnJICrYj8Rz56n4zHvMWLEdURQpLLIx+IkWfDLgce182GXKeHtQr1I4K3YcZtjUxQydski/Fyz78EVeeaIFjzer5SDDMhNf6caz7RvqU+KqqhKflsVT78/CzcXEm8+0RxQFXpown7wCKz+8+Swfv/gYyRkWWrz6JSv/PEKj6lEIAhQW2TEYRN7q3R6rzU7LVyfSbsRXyLKCXVZoUfs+dkx9HReTAR8PN7o+WIsJg54gt8DKxEW/6+dEUZQSP6oCvNypERWi/ShSNV/hCqFl+ObXrfh4uOHuYiYmOYPnOzamTb3KmpZWVpBv0gzexbgWP+5rvQ8vXLiQDh066K+bNGnCwoULr7rO+vXrWb9+PSdOnPhXfLrvKAJ7Nf/BayWwGRkZJCQklLq0/ne4Geb+f7d+UVERMTEx2Gw2IiIi9Fz4e4Th2nC1C9xpLO2UkxQXyDtTUmJjY0lPT6ewsPBvv7P/YgX23nj8b0BVVc6ePUtaWpruAALXn4p4cYNRvUphpKSf51yWBUVVyckvXoE9f4kW0Im+HZtQPSoEAe29k45qaYUQf2x2mbzCIp20iqJIVq5GUm2yTIC3RmDzrVoz07gftDQ4SRS5/4VPiE5OR5YV3nm2Az/+7xlczEaOxWlNMr3bPYBBkhj19TKS07MZ/mQbKoUF6tXc43EpdHlrOtYiOys/eZkXH2lKTHI6b878lZ9/34cKfPPrNvp+8hNWBwF66bFmJC58D6NBYsfRaADaN6hGSqaF3ScTiEvJwt3VTL61iCY1yrNj6uvMHPm0Tobb1a9Kp8Y16PnedxglCRejianLtjB44kLcXUxs++p17gstw6zVf9Jg4HiWbj2op5hNHNyVD/t3xsvdlanLtlBYZKdVnYoUFNmo1e9jXp+6RLNsBT4d+DgHZo7EIInM37gbgBE922qEesJ8UrNyGNOnPW4uJl6d/At1+3/C7hPxvNCxCZ5uLrz//WoORp/l9Z5teLJ1vRKxuJIk0mroRCRRRBJFXEwGElKzmPnrdhRFJa+wiA4NqxE99x0ig/35Yd1OzmXn8lz7RkiSSK8P5vDQ8CnIioLNLhPs78W0oU/i7+WOh6uZTwd20ceOKIp6U9nFUbmiILBk8wFcTEYiy/qTX1jE6F7tmLlimy7fWL7tEILAdc3alQalifS+npmwhIQEPa7bCWec95EjR664nre3d6m9nf8J3FEE9kq4liYuVVVJSUkhOTkZLy8vIiMjS7x3vdOfN+KEUJoHfmFhIdHR0ciyTGRkZAmpw/WS57utkvpPoLhA3pmSIkmSnix0Ja/Zu0lCcK3pL/cI7N0Pp01WVlYWZcqUITQ09IZjvS93G7Lk5bP/eLTe9GM2asTMxWRAvMwwyyxmvWVyLOvjoVWGRVHEaJBoVDWcgmL62QJH/KzNruBqNjr+plV6Jwzuri/XoEo4yRk5SJLIpgNnaPdAVT0BC6Dbg7WxywpThvUkOMAbgyTx01t98fPU+hMqhJbhi8Fd+XXHIYpsdowGie2HorHLCsN6tkYUBH5ct4vVO49hlCTW7DzGkMm/UOCoOn/U/1E2TRxCoI8Hv24/jKqqDOvREpNBYtCXCxk/fz0VQgJocf99rNt9jKgnx3LwTBLDurVi8btax78kiTzfoRGHY5J5+YsFODg+7i4mktKzCQv0QRAE5m3YQ7C/Ny90aIyqqriYjIQGeBMV7M/Pf+wjK6eAdg9UQRQFHhn9NaO+WaZf9z+MfpbtU14nPMiXb1fuYPHmA/xvxq+6Pdf4gY/TvWUdsvMuyPd6ta1Pr/dncSQmmW4t6iBJIq9PXczZ9GwmvtKNH/73rL6sn5c7h2b/j1mjnsFqs9P8lS94/pMfAc3R4Nl2DZk/ti9PNK/Fkdhk/jhwmha1KiCJIqNmLKdWv4+RFa0hzinlcMJJWhVFKRF4oSgKBdYiOjaqTvTZdLq2qM28DbsID/TVG7vyCq20q1/lpvVPXA6l4Sil7VkojoSEhBLktTiys7Mv+3e4oHtdsGDBVeUG/xTuCgJbWvJY3Fjbz89Pt5xyDjjtAr8+C6AbrcBebf38/Hw9QzwqKkrvoi+Ou4mE3q7H4kxJCQsLo0KFCgQFBV3iNZuamkpeXt4tPYbbKcjgnv717oeqqsTExOga/KCgIM0OyfHdX//Yv/CDzDmNay2ysfOI5tVaMTQAJ+NydzFeRDAcvpuyou/fx0O7Twb5lnww92lXH3dHtz1oqV1wwRMVLpBauOA5+9WQbtSvHEba+Ry6vzebRVsOoCgqgx5vzqyRT1OrQojuF1tkl3ls9DT2nIjTr0+DJBJZ1p+ereppRFkQeKptfSYNfoJWdSpyMiGNEwlpPN60JkaDxIodh5m/cS+y43iWbDlIpdBABEHgkSY1aPdAFZ5u8wDnsnNZu/s44xdscIQliDSrWYFKYYH0fPc7REfVcuexWDqNmoaqqqz+5GU+H9QFSRTJysmnR6u67Jn+BnUrhbF4035e+2oRz3/6k3ZuVZWxfdqz6+vhSKLIr9sPAdCi9n3sPh7PzuPxeHm4Ikkiny/cyNKtBylf1h+AHq3q8mLnJkx8pSv51iL6fPg9iennee/5Tjz0QBU27DnB5ws2EBXsz+5vRlI+2F8/750a16DDG1Ow2uzUcIQahHZ7k97j5qCqKr6ebiSey3IEHRRgMkqs3XWcncfiaF2nEmajgaoRZZk35lle79GavEIrC3/fx/AerZFEkc/mrtU9aJ3ygOJjqngDlXYOjXi6uZCTX4i7i9nh+iAypGtLFEUj+S4m401p3roSSquBdX7m0sJisWCxWC75u7MKezlUr16dnj170r59e3r27El4eDhDhgwp9T5vBu74KFkoXQVWlmXi4+PJy8sjKChInxZ2vueMlL3eB/CNWnldicDm5uYSHx+PwWAgMjIS02WSU66XwPxXK7A343glScLb2xtvb28URSEvL4+cnBwsFsslkX/u7u7/CLG7nSQEqvrfiS/8r0MQBMqUKYOqqiWqNsWDDEq9LYQSr5xQFBVJ1Jqddh4+BQgE+XrqJNPf043iV7GiqogIjjAaFclhpxXo46G7GTjh4WImxN9bf+3laM4pjvLBF2wUnaTG3VWTm/l5avZeAV4eiKLAxn2n6OSIJt128DTlyvji5e7C9sNnyLRo5Pd0YhrlQ8ugyDKSJGGQJN1/9cm2WlPUL5v2A9CjVR1Ai24N9PHA082FiYs2Me7HtdSqEELFcoGU8Xbnh//1RlVh5V9HsMsKHRpWI8jXk992HaNpjfL8+tFAtuw/A2jPp7qVwvDxcKXnu7NY/uEAzEaJ6b9uZeyslYx4si3DurdGUVUORZ8F4I0n2yIrKs2GfEGT6lGMd+hd54/tS6YlDxeTke8dYQr9OjYmw5LHhJ9/p0ZkMF0erMXeU4mUD/bngxc6o6gqu47H8cf+U4yevpx5Y/oiigJNa5Zn3obdCIKAi1mL//1q8SaCA7zp+mBtlnzwIoZi0/prPn2ZXh/MxmgwUGS3U7FcICd/GIvVkQA2bOpi0rJyiJ3/Ljn5hXozlxMJC99DUVRkWeGjl4pJBwQBFUrEyTvh5uaGJEn6+K4cHkR2XgF92jUEtHvfmzOX8dGLjyPL8j/6PC2tBnbDhg2cOnVK16VfDj4+PgwfPhwvL68rVl8vlhUUx8XSgSZNmjB27NirrnOzcUcR2Kv5wF7tpmmz2YiLi6OwsJDQ0NDr9ib8u892swmss7JnNpuJiIjAaDSWet0bwd9dIHcL8b1ZxM/pNevp6YmiKOTn55Oenk5RURHJyckIgqCnqLi7u980o//bjcDeis9yD7cHfHx8LrH4ua77QrHhUlwS4JyWL7QWsffoGf0955bb161QYjPFG8DUYtT2la6tOHe+JClxczHpiVLncwuKSQwE7LKCQRLxcDVTYC3C1WzCy72ko4JBEqkUWobIsn4kZ1h4a9ZKttd5jUKrjd7jZrN1ynB9etzL3QVZVmj3+iRWfjqIKuFl+WjueoZ2a4GL6cL9XFYUXujYmCKbnboVy3E0NoW087kMd0gLNh84DUBEkB+r/jzKsKlLWPZBPyqHB9G5cQ0Offc/jJJIvrWIIZN+IdjPiz8mvkqruhUBaP7KBBa//yKzR/Vm6baDjuMV6fNwQ5ZsOcCJeE0nrCgqb/ftyCNNalAzKoSlWw8SfTadFvdXQJJEvlqyGX8vdz2F67OXn2DIEy0J9PVkypLN2Owyn730OIqi0v+zeRRYizj47WhEUaBh1UjOzH3H0RClMOrrXxnxZFtmvtFLPw+SKDLg0Wa8NmURkijSuUkNREHgyXe/o3XdyrzYuSn7Z/4PURR47M3plPXz4uMXH8XXy40CaxGNq0VSPTIYNxcTI6cvY8Hve4mZ9w55BUVMW76Vt5/tgCQJxCZnEFHWr0Sh4UocIj8/H7PZrBePnJVW5/cmiSLDez6ELCslXI3+CSiK8rfPD1VVadOmDb179yY4OLhU23X205T2PYvFQv369Vm/fr1OZJ3LJSQkUL169VLt90ZxV5RLrkbgrFYr0dHRFBUVERERcVXy6tzG9RCMm01gMzMzSUhIwNXVlaioqCuS1xvB3UJEbyc4vWaLR/55eXlRUFBQwms2Ozv7hrVSt5MP7D0LrXtwygiupQIrFhu7FzfMgFaBtcuyPq6ci3i5mUvct4Ri2yh+O3u2fSOqRJQtIQ/w9bxQlSqugTQZJN2qq0iW2X40HtDG9r7TSSU+d6+29QjwcuNEYhrz39L0malZFvIKrLiZTWTlaFpcTzcX8gqt5OQXsuiPvQiCwOin22KQROZt3MvkJVuAC1XeFzs3xcvNlQ17TwDwcP2qFNns7DoRT7sHKmMyGtiw9ySZOfkE+XkSnZzBx3PX4WYy4uPphiSK9GxVl4IiG3ZZ0cMU+nVqQtexMzAaJLo2r40lr5Dub8/EZv8/e+cdHkd19f/PlO1VvVc3jAvGmGZ6SUKHJIDzkg44hYSSQhLS3ySk8QsvISRvXggkIdW0BEhiEgg4EEwxprg39WJJK62k1Wr7zvz+mLKzsiRLtuXYxud5/Hi1O3Pn3pk7Z75z7vd8T5bHb/8499x8NUPROKd9+k7+9PxbnHhMHQiYCW6ffvdZjMQS3PHHf3LXo88BKs+9sYN/rt/O7OoSAD5x+els/uVtLGqspL0vTN9glLtvvBKbLHHNd37NB7/7IDZZQhQ1nvz7zj1Bk82yXJub7n6YvqERfnrz1Vy6fCGSKCIIAv/3uf/i/r+t1YomiBplcOmcah5/cSOSKDIaTyIgcN+t13DTlWcD0D8U5bNXnYvH6eBHDz2rF4nQzApejZXZyZ6DyWQyD5wac3xLq1aWuCjgobe3Z8bLlM8UB3bBggV70AWMSOpEYPT666/Pi8IaerAHC7zCEQJgDac5dgLGYjGam7UEgIaGBnw+3wQtaBPUmND/aQAbCoXo7u7G6/VSX1+/1/4c6AjsoWCHe/KTsdTjdrspKyujsbExT2u2t7f3sNCanepxjkpovb1sMkWY6fgiYRwAm8lkctqsqRQetxYBTYzhFhr0A+uyqgaOtN9T6Qwuh42rzl5KxgIsqooCeXPakMSyyZIZ6ZVEEZ9F1P7Cr/4SyOnWXnfhyTjtNmRJpLa0AFXVSr6CBoSHonFAi/Ya6glBXy53QZYk3mzq4ju/e5rQUNRsWxJFVFQ+cdlpXHfRKcypKuGVbW0kUhlTf/W5N3eysKGCgMfFo/96kzsffo5+vbrYSDzJt669mJd+9jlkSeSa7/yKlzY18+ELTmH1928wk+G8Ljs3XH4GN/74IdLZLLIk8PRrW2ntCfPV+58kk80iIHD9xcvZ/uBXqSoJ8vt/rieeSvOtj2iasd/81WpuvPsRMlmtAMLvnn6NkqD2jG2sKKbz4W9x/gna8v1lyxfyGR1YPvXqFv7y0maWzq3hE5efgSTlYMhdN16Jx2HXKSQinaFBbrhrFW6nnbU//RzHza4mkUrTGRriGx+5iO5Hvo3f4+Sr9/+Fxdd9j93hYfPl5w9f/yi3vu88Mtks3772Yn71xVyk10ofDIVCdHR0TJiQa1g6nTb9tAEOF9Rr3NxEIkF/f/8e+xxom0kZrauvvprVq1ebf69evZqrr77a/Lujo4NVq1YBGjd2bMGnVatWcf3110/rmPtrhxWFYCKzRhKNz5FIhI6ODmw2G3V1dXuVywqFQqYMzL48hMfrw3T3VxSFnp4e+vv7CQQCeZm9e9v3QAHYo1nkB8bGXg9Da9blclFcXEwqlWJkZIRoNEooFCIUCuFwOPD5fHi93nG5zhMd41AAsEcjsEcN9q8q4nimKCoelwZgVTWfMWvQDDKKgmxGaAUTwMSTKURRRJby52TAm4vA2mVJ668u+K/qyWCiIODVtWUFQaCyKFcVTFEU7DaZrKJwwpwaegcjBDwuk6ogSRLDBoB12Ogf1qKxhkTXc2/s4KwlcyjVNWfdTjupdEZLJBIUbLKEJMt8b6VWytXvdjK7qpiTjqllV1eIztAQ37r2IgRB4E//3sAp8+upKy/k3idf5Pbf/oOOh75lnoPffuXD5jlzOe3mOVNVOGfpPM5ZOs98Zl151lJsskzQ60KWJK746r0smVXNbR94J6Ig8F/nngAqnL1kNuu2tbG1vZc7b3g3siTywz/8k+0dvVxz/gk8v6GJh557g59/dgWKqpJOZ3j/+SeaiXbnn3AMkphLbLOaKAgEfW4URaUzNEh1SQHf/ujF/HtDE2cumQ2qyrOv7+CTd/6RB7/8IU5fPEsrR3vqQra191Ls9/La9nZuvPthHvnmdVSVBFn9yhYu1Is3mPNKVTXeq6oya9YsRkdHiUajDA8PMzQ0hCRJJu3L7Xabfi2RSJBOp3G5XNpzF5VkMkVbW9tBeW5OJ4lruoG4W2+91azENTw8THt7e56269q1a7nvvvtYsWIFACtXruS+++7D7/fT3t5OTU0NK1eunOaI9s8OKwA7GQcWcg/RcDhMd3c3LpeLuro6M0FrMttf7sr+AliAeDxOJBKhsLCQioqKabVzIGW0pqpteqRFfQ+0TRalcjgcOBwOE8xGo1Gi0Sj9/f309/djt9tNKoLD4Ri3rUMJwO7LstVRO/LsQLxMW+eZz+PCqwPYsbPPTBrLKigISJKwByAygJIkCvQNjlBa4DOBC2iRUONwkiCgiEbbGrg1bH5NKQD3r36F6y48mZbdA9SUBnE5bHzzl3/hO9dfRmg4qrcpMjyqAViHzcaQvgxvJH/d8pOHeO6uz/Cxi0/lnj//G5fdxj2Pv8C/Nzbzuy9/MK//qqpy3Kwq1t7zGa0Nv4c/fv3DnLGoEYBPvfsM3nv6cYiCwOLGSrb9+it5+1opGqIoEBoaodDvQZJEbv7JIyxf0KDquFdgAAAgAElEQVTLVmlyYFecvlhLAnp9O2s3tbCzI8RXP/QutrTtJuBx8fHLTiOrKJx4TB333HQll5+2mJbdA7ywsYnff+VDyJLED/7wDPNqSlFVlf974kWeeX0bj31rJaIoMBJL4HPnc4qt2f9mFFpVqS4pMMd8pi6D1dQV4qJTFvDW/bfhcTmQRJGtbT2cv+wYztfL+o7GU1y2fCFVJUGSqTQ1pUEzYc48F0aEP5lEkiQzkclIyDX8cSSiqTpYcxhAS64ee50OBoCdKR1YwyYDoCtWrDDB61S2Pxh2WAHYicy6HNDf308oFMLr9VJbWzvlizh2Ukw3krA/oE5RFDJ66bmSkhJKS0undTPMJIVAkiQkSSKdTh8xgPVgUgimYobWbGFhIel02nSe4XCYcDiMzWYzwazT6dxjrh0MDuxUJLTgKIB9u9tU/GYqlTL9yt4s4HXj1QGPNs/3nOuKqhKJxSnwebDLssmBNeS4ZEkCBHZ0hkxFApMuJubaFEUBVcnta7Pl+je7SlMl+OsrW1nUUMGSWZXIkqRxW//1Oj/4xLsJDeoRWFFkJKYFRGRZJKIndAV9brJZhe7+Yb7z4N/40aeu5OkffgJRFNje0ceat5rY0RliQX05bT1hVr+6hU9cdjqtuwdQgYaKIoZH45x7/FyzX+8/b5n5efnCxrzzMtYvhIai5hI/wI9ueA+KqlWeuubbv+Kk+XV8bsV5qCqct3QeP//sCgIeLRr7uZ/9iTd2ddL98O2MxlMk0xmu1hO56ssL6X7k2yYIfegbH8XvdqIoGhXi45eeZh5zLHg1zpf1886OPmbpvFrj+hrl0iuLA7y4sUmrwAV09w9z3Q9/z3euv4Rzj59LIpXmrCWzOWOxluTnsNtY2FCZa8viyzTFivy5OjYhNx6Pm/54ZGTEpIX5fL68hNyDBWCnwoF9O/niw26E400S40L19PQQCoUIBoPU1dVN6wIKglZpybDpht/3FcAqikJ7ezuKouBwOExNxekeeyYKGdjtdlwuF7JNiwa6XK4DlkF/pNu+gn2bzUZBQQE1NTU0NjZSVlaG3W5ncHCQjo6OPK3Zg8U7nc6y1VH6ydvDJltd2BuAHQ84QG4OWW+dgM9jUghEMZfENfaYxpK9TZbySoEa/FZBgFOOrdtjXytnVqvCpH8WBOwWX1dTovH9+oai3PP4iyaF4LXtbRT7NSqAEXUVRYGEXihBEkUiOpj1uZ3m543N3QiCwOwqDai19w0BUBzQInx15YXUlAT58SPPUV9RRG1pkJFYkvkfvt30LZd++d489QXDsno51O0dvZxx011E40m2tvWw4Nrv8crWNrKKwqfvfpinX9uKLEnYbTJ333wlqXRWK62rJ5defvpizlk6l52dfazf0cE15y1DFAV+9NCzLPjod0mk0vQNjXDnw8/pkWyBl7e0EvBoy+uv7WhHFHVqhqWfRtKWokuejfWVc2pKTbmJr97/F9Zvb0dRtG1dDjunLGhAEkV2DwxTXuRn7U8/a4L6r93/F4ajce0aJFOmvJ9hz76x0ywjvLckWiPyauQwVFdXEwgESCaTeQm5Q0NDU4qM7q8ZY5nJCOzhZkfUCCORiFkVZrqTSRRFM4vO+Hs6ti8aiJlMxhQEt9ls+6w0cCAjsMbDxe32YLfbEQSBrbu1ShyCKOJ2u6moqKSwsPCAHO9Itv11aLIsm1zoWbNmUV5ejsvlIhKJ0NXVRV+fJn0Ti8VmNPv1qNM8alO1/dHDTmWye3wXHo6a+qsT8fNtsmSqCUiiaCZsCUAqlStSI4/z8p0X/RvTvpWOUOzXErCGonH+sX6H2eabOztx2LWFzHQmiyzr1ciM+0EQSOp987ocRHSQO6sypzML0BnSAGyhz82q597gb69s4cJTFnDRKVpGtyRJ+NwOls2rNfv4+RXnmp+tz4AtLT1IoiZ5JQoCXpeDx/6tSWctaqjg5S2tPLTmTULDo4iCwIP/eBWP08EXr3kHIHDWzXex4psPMBJLIgoCc6pL6X7kO/zg45cD8Nun13HLezUZsLseWcOzr+8gqyj89eVNbGvrMcd04rzcC4NxPgRB4K+vbEZRFMKRGCv++5d55/x93/oViqKyvbOXRCrNt6+9mGXH1CGKArFkSovG6nOsqqRgj3j8HZ98t8lxdjrsqGouEt/U1c89f34ehy6hNlYGbjIzIq+lpaU0NDRQW1trrpj19fWRSCRIJpP7nJA7HZtqMOHt4IsPewpBJpMhFAoBUFxcTFlZ2T61M5XIwd72h6lH3tLpNK2traRSKWpqahgYGDjoS/Rj+2y84RkixKmMgl0WOLZS03dLZhQcsoAsS9TX1yMIAslk8rCjFxxqFIKp2FieViwWIxwOk0gk6OvrIxQKzYjWLMxc9ZejduSZIAhTAgbj+Yvxovg/eOBRFs+tB0AUxp9bkiiakVYEzEQsQRCIjMbwe115XFCriaJgRvtEUUBQxgewRX4NFAW9LoJ64QNREGjtGeDcEzTu5dc+fBFXnLEEgEWNlRb/qrXhdtjo6BsEoFavVDUaT+K028hkFeZWlWCTJd5q6uKPz73Bsz+6gVmVJby2vZ3ugSEuW76YJ27/mNmnMxfntHAVRUVRNbpEQ2Uh8WSK9565hHfoffvn+h00VhThdtr511tacYNzlswhNBTl8z9/nP97ci3P//gmJFHkn3fexAe+8yszUqmdC8m8Zrt+93Xz8+dXnEeB10VWUbn4lIV50W8ELRqsKCqJVAaf20FH3yDVxUFEUWT3wBDvO3dp3vX45kcuJKsorPjvX5LNKrx4z2cIet06zeGX3HLluZx3wry857UBaie6vrFkCrfDzupXt/C1D7zL/G1fZQwFQcDpdOJ0OikqKiKVStHZ2YmqqnkJuQbtywgE7a9Np6Q37Jua0uFmh/XTJpVK0dzcbL7xTCaTtTcz3mCtIHY6D+PplFG09ruuro5AILBfUdQDEYE1uDVG5Q5VVRlNps1sTQC7JJoOyriJbPaj9IKDbYbWrDHfKyoq8rRmm5ub6erqYnh4eFpRholsphMHjtqRY1ONwI4LYC0vloYfttlkfO5coYGJzEy4UlWTJiuKgplYlR4nugt6Mo++vSDkqxzIliSuIj0B64X/+RQPfum/zO9/fNPVfOMjFwNaRa/TFmmg8uJTF9Hy0Hew22StbCxapNjQma3WKQmf/dljSJLI3Z9+N2cep+27q6ufaDzJ6zu7EEWBudUlXHrqonGVTXKWA3Fel5O+wRHOXjKHS05dwOBIjE2tu7nmvBMAWLu5GYDyQj/Pb9AKJHzg/GVIosiPH/0Xiqry6LdWUlbg4+5H1/DnF94yj6foL7OiKPL6zg4KfW49uq0d2wokjd7Jkmieg5rSApbMrgJgUWMV7z1rSd6Y5lSX8MSLG+gJR5hfV0bA4+Jfb+4kmcrw6LdW8tdXNpnPIFEU8/xNdoIAlENP4v70u89k6VxNtzSb3VN2c1/MSMiVJAm32019fb1Z5XNgYIC2tjZaW1vp7+8nkUjs1zGn6mPfTkVlDrunjXFR4vE4TU1NZLNZysvLgf3LijfatdYxnop6wdj999aHRCJhatPW19ebgvf/CQBr7bMky9jtdrKKSksoSjyVwe+yI+pOa0PHIKIoaLIzqkr3UFzL7DSWsETtBvZ4vTNSdOFws4NB6jeuuZWnVVNTQzAYJJlM0tvbS1NT035pzRrHmQnx7KN2+NpkijD7upJlyCyJYi6KWxTwmRQCwx5buzXvb2M5GbRgquGTZEmis1fLa5BEgV2dvXn7GOOwjiSPQmCZywFPLpGssaJo3P4PREbNz8l0Bo9T6/dV55zAe8463lzKBg08Avx93Raefm0rZyxqZLnO0d3Zpa0onnXcLGLJFH6dUyoIArFE0jzGl+77cx6osT4D6vQIryiK7OjsQxQFTj22gUxW4a2mbs5cPAubLLF2cysA71g2j5FYgu/9/mnO/cw9ZHVd9erSIB+/84/84q9rtfaMhGlFYemc/FKiY00QBJ57YyfpjFbd7Lb7nuTHj6xBFEW+//unOfuWu/cAnZIo8u4zjuNfd93M/3zqPaQyWW74n4e47Cv/RzyZ5o5PvDsPJGv91DR/h0ZiesRX4bk3dnD1N++ntWcASRL59oOr+dzPHrOsNh5YypXh742E3NraWhobGyktLcVmsxEOh2lvb9+r1uzejgFTj8C+HXzxYTnCkZERWlpaEEWRxsZG3G6Nn7Q/FADjYlvbmK4SAEwOYI3CCgANDQ1mv439/xPL8G63m+KSUkRRIqMorG0K0VDixWGTkUSBl5r6EQWBY6sCqCqkswprtvZREXDqFVFUXmrqRxL1t3Mw9UwNrbxDzQ5HCsFEx4DcOARB05otKSnJ42kZYt0tLS20t7cTDofzXtSmcpypOs1D8XoftYNn++PHrL7XeNnSAKwzr82R+J5z1+TJAi6Hpt8qiSJbWrrMfs2uLht/Gdby2fq9ZIn4Bjx7Zs+PNSu47AtH8n7738/+Fz63E6/ez+KAl0QqzWg8xQ//8AyyJPHOZceQSKXpHojgdzsp8Lp48O+v8vt/vma2Y7cEB76/8oq8/k4EWE6eX8/mB25j2bwamrr7SaYzXKTror62Xas2VlNawPMbmlAUlfJCH3abTP/wKO85YwkP//d1NI7h7A6OxMwXDsgljqmqalYh++3Tr/Kx//d7BAFe3tLC/X97iXedNJ9kOsMv/voSx8+p3mPpv6NvkN8/u5451SVUFQcZGI4yOBpjQ1M3bb1hJFEkm83yxIsbtTELAms3NWG3yRTolchA4MzFs7nk1IVUFQdZu6mZn/zpeQp8OT3X/a2CONbG85GyLBMMBqmurjZzGBwOh1ki3pqQO51Vi6N0rpwddiMcHBykra0Nu91OY2NjnkbmgeKw7ks7ewOwBuiWZZnGxkacznyH+J+IwDocDubPn49NlmgfGEUWRc6Yo+kd9o8kSKaznDqrGFVVkUWRrbuHsUki58zXZL7a+qP0jiQ4dVYxWUVBFAT+tnF3bqlJlEwh6KNR2QNvk11zg6dVXFxMfX09dXV1FBUVoaoq/f39tLa2mktbyWRy0ramA2CP0kje3jbVCKy1ZKVhVkBk6HIXBnx4dUkmw4bjyXH23ROYCqLA+m0ted8nk3vuK0zw2cqbddrHX42zjtVQQgCIxnO64u29AyaYWDavjrk1pfjdTlO14M1dnQDYbTIdoSFUVeV95xyPKIr8e2Mzt937pKVPY45vuW8Nfu145nFqPMx5NaW88rPPcu2Fp5BIpdnRGeK42VU4bDIvbtLO1Q1XnAHAJV/+P/747HrOXDybc5bM4U8vvMU9f/oXAP3Do6zb1qarAyj0D43wh3+uZ2g0jl+nfFxz3ols+83XsMkSrT2DlAS8zK4q4cm1G4nEEnz4XSeRsVA7VFXloTVvYJMkZElCFEUqi4Ns/uVXeP/5J7CwoZJwZBRJkrjstEWAdl2XL9SoFyoaID9+5fdZv7ODD73rZGyyxB2rntHGc0quxOmBoFZZbW8+0shhMBJyKyoqcLvdjIyM0NXVRVNTE7t372ZkZGTC+2e6HNijAPYQM+Ph6/F4aGhoMEHRdPinE5kVBFuXWqdSEcm6/3h9MKpaOBwOGhsbJ2zzQPBjpmpjVQ9qCt15v5f6ndikXGQPYH5FwPxbUVTqir2U+zUg3hfRHgwXL9Y090RBYEPnkL69iNPpxOfzHTBC+6Fu/4kI7GTmcDgoKiqirq6OhoYGSkpKkCSJcDhs8rRCoRDxeHyPuTQVDuxRGa2jBlN7mTaWU8da1rKfMQ8LA168bhcqlgjsaD4InSgBVxQEmjr6cu0rCjt37tyjaE1+NDbvB/Ojofk6VvbJCoRCwzkKQTSeeylMpNK5JDPg6R/dTG15Iam0tu+C+grzt0qdWvCuE+ejqCrJdIZ4Kvc8GntmjR4mUmkzuz6TzfJWU1fedsZv8WSaQq8bVVWxyRI/vflKbr3qHCAXjV2+oJ5t7b207B7g5nseY3g0bkp+fevXq7n3yRc5praME4+pRRQ1PmxRwEc0niCZypBRFG788cP87+MvmBHW9527lA0PfAlZEikv9HNsfTnz68pNuoRxHf768mbOWTIHgK7QEDff8yguh43/98n3kM5kOPOm/zErngGsfmULsiSSSmd49vXtnDC3hneccAwfuv1BMvq1+ektV+N3O5lbW2aenwOt2jIdf29ozVZUVNDY2EhlZSVer5fR0VF2795NU1MT3d3dRCKRvPl1lAO7px1WAFYQBBobG6mrq8uL9ByICOxEIHiqk2AiABsOh+no6MDlctHQ0DAhr3Z/I7BTNVEUcTgc5njDUe1hYJDws4pK+8AoiprjVfVGtEiBsaSWVVT6o0mT0A9Q5HXkRQMAjqvWEhUyikI8pd2I9kMg6etgUQgOhu3LGMZqzRo8rbFaswZPazri2UcjsG8Pm4wDO1Z70zBVVenr66O7uxufz7fHNlkLyIvH42QVhaKAj8KAFytTNZnJmhnyqqrmLQePVUHoCFm0vfXo8HhRWHN/61gsf4zGk9z58HOIojDh8rMo5MYzmkiZ/jCVzppjHY7GcDvt2GWZmlKNt3vqsQ3mfk67jYDHyfy6MkRB4OFvXss/7rjB/P0dt/5v7hxZaEDhkbgp55XNKlQUaUBYGePrHHaZeR/5nnk+Llu+iHeeOB+AudUllBZ48btdPP3aNgBOX9hAwOOiubufRY2V/P2OT/G7Z9aZBQHWbWvjhI/9gLeaOvnEZWdQXujnzlX/ZNVzr2vRUlHkS/c+wYe/9xtS+jU7fdEsnrvzRhw2mYA3txKpqCqbWnZTpGvhrtvexh/+uZ4v/PxxRFFAkiROmFdrFmNQFIWywlzi9qfvepiOvkFuX3kp//uZq5Elid89s47KoiBv/eJLuHVqiToDkoP7GrAwEnLLy8uZNWuWqTWbSCT20Jo15t1UIrBvh+grHGYAFjReydiLcyAjsEYb011iGA/AhkIhuru78Xq91NfXT/pw3x8wNVXwK8syNpsNRVUZiqXYunuYoFu7qRVVZe2uEJIoUF2gRWNHEmnWNvVT4tOcjKKovLhT26bYp3G5ook0m7qGkUStBnlWUWkORfWbSBuT9n++I5VlGbfbjfcITvo6GBHY/T3GeDwtQ2u2s7OTpqYmMpmMWSluIns78a6O2sQ2UTBBVVV2795NX18fwWCQ2tracSL9ub8NjnZhwEdR0JenQKCoqqkqkM1myWazE0ZFM5l8Xm1DQwNOpzMPLI8ZgflJFHOqK/f+dS3vO2cp2aySNzbrcRc1VpufRxMp8zfrMnkskcpLXLrq7OM5bo6WlX/j3Q8jSSIv3n2zWdAAYMnsXLvtPTmaQFNXDpwPR+MEdHkvh91GkU9fUVPzz6soCPzic1cD8NSrW1l03fdJZTJkFYV7br6KTQ/chigKZjT2tve/k6yicMXX7uP+v73E8XNqWHPXzWQUle7+YZbNq2XlJcu5+SePmFHmW646l4tPOZaPXXYa4cgov316HWs3t2C3yTz491c5+5a7ad2t9b2yKJh3LquLA2bUtnl3GIAPveskVFUlPDLKL259f945XtRQSSarYLfJfHflpXzyzj9ikyXOWjKXFzc28Zl7HuXTdz2Ey2Ezz8NMaGYfCF88Vmu2pqaGgoICU2t29+7dgFbGdrKE3INRVOFQsSPiaTNeAtb+tmF9y54KwLICWFVV6enpobe3l0AgMKWStjPJgRUEAUm2ga6juGZbL0G3nbllPgQBdvVGGIqlWD67ROOyigIvN/Xjc9o4tbFYW4rrG6FvJMFpc0rIZDW+69837cbrtHFshR8BaB8YZWPnEHVFHpMH++zWXmRRxKVHB4ZiqbwkDEEQsOtJX06n84gBQAeLQjATWrOVlZUmT8vj8aCqKvF4nObm5gl5Wm+nZaujNrGNF0xQFIWOjg7C4TDFxcVmoZmxD+Gsmg8MRVGkKOiltCCQN680ab+JgxZWfu25y441P0uSZOYeTDRNrUB5NJFi4bXfJ5FK894zlzCvtoyxi/jW4xf6c6BTySqm/qu2UmX5bAHJ99yygstOWwzAY8+/ybptbXntQG5pH+D0xblo7aMvbDA/b2nvNe/JeDKNAcS1Jf78wV5y6kIEQeC8pXMZjCaQBJEH/7GOux5ZYwL7+79wDQ9/81oWz6rkn+u30xOO8OVfPElH36CW/9DTz47OXiKjCT52yek8d+dNqKrKz594AZsscv8X3k9p0Mcz67fjctj49BVnIokiT760iS1tPQR9bra19/Kzx18w+yWJIq/d+4W8c+t22jluVhVPvbqV93z1F+ZKoaqqfO2Bv2KTJWRJJJtVuOy0xezsDPG3lzcjCLDmzZ0APLTmdf6+bqsZjZ4J/uuBNmtCrpHDYCgWDQ0N0dLSQltbGwMDA3kJuW8n/iscIQB2qhJW021jOpPB2D+bzdLd3U1/fz+FhYVUV1dPef+ZuBFkWUaSbciSyCadk3rufE12TBJF4ukss8v8FOiRWGOp/7Q5WolDURRIZRRmlfoo0/mugzHthrlwUaW5TedQjNoiD8fVBJFEgXUtA6QyCmfPKyWrv/k+sq6doNuOz6XJdW3sHGI0mTHduSjJeDweGmfNorg4P+v1QNqRQiGYSZBs5WkBpu5sLBYzeVpWrdmjFIK3n40398ZGYLPZLK2trUQiEcrLyykvL8/bz3qfZMbw/URBoKQgQFEgX99bUVQkKf84E0VgH/7eTaQzWdLpdB5gtm6z5q1d9FhUA97cpfFHv/Gr1QxEYsSTaWpLC8gqyqTZ61ZtUSuf10q1skZDRxNJUDGXttOZLHc+/CyylC+J1RnKRV0XN1aan512W+65o+RYwlvaeglHY2afIqP5nF+jbZsscfVZS5AkkTd2dfHd3z/DaELz7Wve3MVZx81GFiUaK4uZW13CBSfNp7okiKqqzKku4+wlcwl4XYiigE1XrfnIBSebCViCIHD1OUvZ/Msvc8XpixmOxnlxUzOlQS8Bj0ZTiCUsNIjIaF7i3I3vPpM/fft6bLLEb55ehyjkKqXt6grxxNpNZLKa+oEkidhkifu/+H7OOV7j0N74nrMBbU6ePL8eWdIoJDMFYGcKNBpaswaAraqqytOatSbkbtiwgeHh4bdNIOGwA7AzoT9obdfqOKYjNWTsHw6HGRwcpKSkhIqKimlxaA9kBFYQBGw2G5IkmYDzuNqCvG0UVUV/DuRJMeU9DBSVaDKdB5Z8Ttsex6vwO/PaObGhCIcsIooCbQNacsN7l2m6gaOJDK+2DLCoOojLrtXQfmZLjynFJQoClZWVh7QU11TscIvATmZ2u33cmuC9vb3ccsstfPCDH+Tee+9l7dq107pvjtqRZdYIbDqdprm5mXg8TnV19bgvpeMlqVittDCAz+MeE1iAzBgh+rERX8NGR0dJp5J7PBs0IKOBn52d/az80cMAfOZnf+byr9+PqqqUBjXAEEtqwHeial6GDQ8PmVFXmySaUVdVUc2XdFVVTQpBJquQtIDqK89awj/XbyeZyuTd17PKc7qztaU5H15u4X+6LGDWJks4jXKpqsrqddvM7eLJdF7bH7vkFAA2NncT8GgyX/f86Xmuuf1BdnWFUFWVxspi/vXjW3jg1mvMxDLQtFer3v1FfvLoc9r5EUVsUi7Hw7gmdptMbVkBO7tCnLFoFtdedAqiKPDsGzux23Lbjyby/UYmq7BkVhXZrMJ7zzyOn33mavPcrX5lCw984RokUTApB2/t6mL5gkY8Tge/+ccr+D1O/vujF7OosdKMak9MHTn0zTyfFq3ZhoYGSktLkWWZDRs2cN1113HDDTfw9a9/nV//+tf09vbupdXD2w47ADuR7W8Ec280hKmqESQSCcrLyykrK5sWuDiQAFbWixIYYypw2/Nu3Kyi0tof1TRbbbLGLxpN0jYwitOmAcqsRd+1wG1HBYZjKTZ3DenOWTCjqKA5TdDKz4ZGEnngqjLoyvvb45Q5pVFzyruH4owk0px/bDlZvTDC42927SHF5fP5NP7uDPCXZsIORwrBRMeA/BecsTXBa2triUajrFmzhptuuonly5fzyiuv5LVz33338dRTT/HUU09x3333TXrMtWvXcscdd9DR0QHAqlWreOqpp2ZgdEftQJvhc4yCLUa1wWAwOO72BocVAHVPPdbioF8r8TqGQvDshta8aKbVRkdzagCiKE64KiAIGhiuKg6wvVNTKygJekmkMqSzillsIJ5MmX2dzOLxOAbFYHZ1qaW/irkKZeXvqopKZDQnvfWzz7yPbQ9+zUzGMqyyOHfurElLxQGPeX867bIJku02CZueLCwAlUUBc5/lN/8EgK7+YQAWNlSSVRTu//z7WHHOEo0+trUNWRapKytk9atbOP3Gu0hnskiSRE84wvJP/pCmrhDhkRipTBa/WwtevPsrP6epO4SiKDyzfkdeMp8gCCydW8Oqb3yUa85bRjyZ4tVtbTmuLuwx7g3N3ahoig5XnnU8CxoqTLB6wUnHctZxs/Okw9p7w0iSSGhohFt//me2t/fy0QtP4cKTcjQSRTmw0Vc4ePSp8Y5js9nMHIZTTz2VK6+8kmAwyJYtW/jud7/LhRdemHc/TGZr167lpptumtK20/HnM2lHDICdiQgsTE0eKJPJmLIwfr9/n5a/DwSFwFAYMBx2JquBSQBJEskqKlu7h8kqCjWFHjNa8NSmbgo9DqoL3IgCbN09TGgkwcmNRWSyGiF89aZuAm478ysCCAI090XZtjvCgsqAyS16/I1O7LJIsdeBokJnOMbO3pE8wNvcH81bUisPuHDZtf4ab9dXHK8lLYiCwNpd/YC29OZwOAgEAqaKwr5e7yOFr/mfluoytGZvvfVWHnjgAT75yU9y7rnnYrfbiUaj5nZ33HEHfr+fCy64gAsuuICamhruuOOOCY8ZiUT4xS9+wds91HEAACAASURBVPnnn8+8efNob2/nggsuOPCDO2oH3Ix50tXVhaIoNDQ0mEufE5kxx6wAZqw/zIuwqipOu7yHpJRhw8PD5ufJlnUVRUESBWpKgwyPJshkFTPqms5kKSvwaStZsqRFa8eJFltpCYFADijOqy03I7aiKJr0CFVVSeiyWCqwWweSiqKgKCqFfg/RMTq3XneuElmxP3cuC7w58Ody5CKwdllCsgy7JJjj1B5TowHrquIATd1GlTKRxspiPnnp6QC8uq2ND7/zJGyyxOMvbuQ3t30Ap91GNquwsamLjr5BfG4HYb3yWEmB1qeNTV18+PZfIYoi558wF1EUeehfb7Fbp2cY56Mk6GVHRx/pTJbFs6rMvhlA2LDZVSWIgsDlX9UAktUHza4q4ddPvcIHv/tr87tLT1tEVlHw6brB3/7Napx2Gze+5yzz2XKg6QNw8J4ne8MiLpeLz372s/zkJz/hhz/8Iddffz1XXXUVDodj3O0N27x5M/fddx9//OMfGRkZmXRbmL4/n0k7YgDsRFqA09kf9gSwVgc13kRIp9O0tLSY0iwul+uAHn8qZrPZWLBggZlsllVUdvZGkCUNTIKWPLWpc5D5lQFEQUASBda3aVmeFy7SnEhGURiKpZlfEaDM70QUBDrCGpfqYn0bURToH0nSWOrl2EotOrKxY4isonL58dUoimpSAqoL3cwq9SIK8HpbmK7BGHWFHgR9nP/a3ocg5Eo2xlPZPWqWnza7WO+bSsiQ/JJteDwefH4/kiTpzv/QisweKRHYqeq7er1ezjjjDO6++27Wrl3LeeedZ/720EMPceGFF5p/L1++nIceemjS9p555hmeeeYZtm/fzq233rofIzhqM2XjzQlDY9WokjhVf6goKk57Lll2UgCrqFQW+vIqZVl/t1Y4nMyy2SyCIFBTokU405ksJTqATaYzVBT5ueDEY6grK2Rs6dFUKmVGmU0Abnk+GDJTADZZNiWkBEFgJJbQP8Pu8JD5vSBAKp3GZbeZIFfbPxdB9lvArNeVky70unIrhDZJyrs2Pksp3vl1Zebn5Tf/JE8RobzIz8BwlKFonA+98yRSmSx3feo9zKoq4Z7H1qCoKn2DGsBx2m2mHmuhz0NWUYjEEnicuX6oqsojz7/Fko/9P3P82axCLJFiYWMlr/388zRY6BF2WaK1J2yOKeBxsq29l1uuPBuA0GAOXImiwLJjagl43JYIucK9T/xbB61n8/dXt7KxuQtJEhHQ5tPhDGCnwrU15t38+fO59dZbue222yaU7jRswYIFrFy5ktNOO21K/dgXfz5TdtgB2L3pD+5vu+MBIWu71uWoZDJpLpPV1tbuse2+2HT3l2WZiooKTRpGUdnVG0ESBRpLfGR1MLm5e4ig287iGo0/NRzXnOOJDfk1ve2SiNehjc84H9UFY/lnKgGXLW+b42oLTL5XWj9/71qoJf8k0woDoymW1hVSXehGEgWe29aHIAicPqcEVYVYKsNj6zvwu2xatMNCTTAyaCVRwOfIPeAUnW7gcrkIBAI4nU7TQR0KYPZIAbBTTVCYSEaro6ODSCSC3+83v/P7/UQiETZv3jxhe4FAgJqayWutH7VDywYHB03OXVlZ2V4jP4YZS81GCVjYMyBhzENVVVFUDcAa3xkqB4ZZ59pULOh1EfA4SWezlOkANhpPUlnk59YV52qJQmOATzKZZPv27RMCIkXJcXTry4tMCoEkCoQj2upEwOPinScuNMcbjUbZtnUrsdhoHpi33lNuC0D0Ou2mKILbcu5kWczzC9a2Givyfb5BLcsqColUmqKAl+bffZ35tWXYJBGnw8aTL27g9t+uRpZE+nTQapMlMwIb8LoZiWm0sXodkHaFNP9dU6JHpQWB0URKp22oSKJIbVkhbqdF4UcQePKlTWafjJeYy09bxEBklJbecN75n19bzhPf/XiudLAk8ujzbzI4EmPlJcsBeOi5N5BEERU1T87sQNrBBrCT2UzLGe6rP58pO+wA7ES2vxHYybRkrU7KeJtJJBK0tLSgKAr19fX4fL4J95+KTXfyi6KI3W7XIpCqykA0SWt/lNllfjJZbWmsXU+eWlxdkHcMv8uWp4+YVVRaQlFUcm/7g7EU3UMxk3+mgUpNQsVh07aJpTJ0hkdN7qqqqgzH0nnnwCGL+BzaOTOWkM6dr0UBJFFgR+8IbrvMFUurEQSBrsEYL+4KsbAqYD7YVm/cjSQKJtUglVUY0hPTjDE5HA78fj/BYBCbzUYmk5kQzB6lEEzvGLDv5Qs7OjomBBTWpd6xZvBeV61a9R9bnjpqU7f+/n66urrMiOt0H6Aqah44E0WRTCZDMpnM0yAWBAG7LFHgzb2wtrW15UlnTefYxgrbh96xjHgybfJeR0aTFAe8LGyogDHRV2vktaGhIe+38aLGfo/TTCIqLwxwbH2lORbjBV1VVZqamlBVdVLNZYcl6cnlsJk0CitIlSUpryCD4a8BygtyFAS/24FNlnhozZvEEmlQYXt7Ly6dLiAIWoLUS5ubmV1ZgiAIZhRUliTCEW11zud2MDiifa4pKwTgi//3Z4ZH43z5mndwbF0Zdlniu79/mi/e9xf8nlxU3nqtREHgb69sIWMZ+/w6TbXi3r+8RJlewCCVzjA0NER7e9seqhDf//jl/O7pdZQV+llQX8HyhY0mIB4bRT9QdjABrCAIkx5nphUR9tWfz5QdMQD2QEVg9wZgQQNLLS0tADQ0NJhLVvubiDXR8a0miqJZBtY43httYYq8DhpKNOc0ktCccm2RJ6+9VCZLVziGKAjIsibS/VbHIJmsQm2Rx3ybf+LNTgrcdioCLgQBtnYP0z4wyoLKoJk48ehrHbjtMlV60YNdfSO82T5Ikdehv0yo/Gt7L6KYA7zRRJr+kWRenxqLNccuCppcV1WBmzPmlOiR42EGYykuXFRhLnX99qU2nDaJQq8DRVHpHooTGknkVQFzulwUFBTg8Wh6tOl0+qBHZo+0COxUAezY7SKRyLh174239vFswYIFrFixggsuuIAVK1ZQW1s75eSCo3ZwTVW1AgU9PT2mhjDsv1i8leM+9t4t1TmdiqLQ2trK6OgoVVU5LuV07glF0ZQIPnX5aSRSaZNCkNCBrTJm2TkWi9HS0oIgCONSJKyRYmuE2Oi/zSabQQJVVU3qmQFM6urqcDgcJBKJcSW7ZAudwG6TMJy205773iaJeQBGthBiCyxJU5eefCyCIPD0+u3c/NPH8LgczKkp1SS4LKewtWeAuXpSWt/QCHZZRpZEwiNagMTjtDOgl9E1Es6au/u5+5E1FAc83HjFGQBsbevl1/9Yx2vbO1BVlRc3NpncYEVV6eofZv2OzlwE1rhGqsofnl1PoV/ruyhoNJHi4mJisViejN+S2TX87plXEQWB26+/lPNPmIcgaDzfmaAPwMHlwO7tGDMdgd0Xfz6TNjk54jAyURQnrU6xN5uMQgB7goUlS5YwMjKSp04wkwBWEIS8KmSKorKpa5DFNYUsrdPeeo0Ip1dflrGWfn2tZYCTZxVTUaApAvRGkvQNJziupsCM2K7dFWL57BIuPU57GMTSWYZjaeZXBszxv94aZml9Ie9dVmN+91Z7mKV1hdqyvijwSvMAc0u9nDFH04CVRIHfv9LKNSfX43Vq/dnRE8Ftl8yqX1lFZfdwjJoCjxmVWFiVA8y9kSSVQRcfOLXOPO5jr3dypd4PRYV1rWFqi1yU6pXDREnG7/driROJBIlE4qAB2f2lkkz1GIcKB1ZVxy9fONHb+thlKKuNpQ4sX76cr3/965Puc9QOvqmqSmdnJ8PDwxQWFlJRUWGCrunMf0VRkMV8pYDJfGm5DmD7+/tJJBLU1tbmzYvpPryVbJaA12WCO6/LYUY0sxYQOTo6SltbG5Ik0dDQMKkyzXjJwAYoz2azZh+j0ahZXKG+vj6vaE4qlUIaw2eVLJ/tsmRGYG0WnqOhkWuYUfQBoNAS/Tx5vkZ7a+kJs6G5m11d/cyuKqbQ66ZvcIS1m5p571nH07J7gEtOXQRAaGiEmtIggiCYUVe7LJtgtqxAi5IODEd5YPVLfOkD7+SK07V9W3UKQGmBF0EQWDavlu3tfSxoqEAAHn3+LUDjH3tdDj3xWGHdtnZ6wiPmNbGeW0EQiMfjZqKgLIm8Y9kxvLq1lVOObUCSRNbvaOf42dUkEom98kH3xQ5mBHZvc/vJJ5/k3//+Nw6HY0IueDAY5POf//w++dJ98eczaYcdgJ1okuxvFv/eAGQqldqD0+X1evM0L2cCwI4FrllFpWtQKxqwuKYwb99MViGWyuBx5JK5NnUNcVxNAcsaikww2dwfpbHER6leDjaSSFPocbB8dkleex67jEMW8747vq5gj6WMJTU5bcKMonByY5G5zbbdEY6p8PNfJ9WZfVy7q58z55Wamq9rtvWxsNpPdYEHQdC2ebV1gJMbik0QHnDZNMFqMcfvulLXlY0mMzT1RTmxodBUOPjbxm7OnluK0y4hCgJ2hxOXy0VWUYjH40iSRDqdNgW3Z+KN9UiKwO7t/EzkXK2Z2VP5LRKJcOKJJ/LMM8+YQNbYrqOjgwULFky570dt5kxVVXPpvrS0lJKSkrz7aLoviWOn8WQravOrtcTOSCSSV6Eo19b07glFURAVBfT8hjfv/Tw+lyPvZXdkZIT29nbsdvseQNMwa38TiQQul2tcCS/rdu3t7RQUFJDJZMZtMx6P5wERa1UtWZLMVTObJcoqWQCrIGgRS8N8nlym/6wq7TyeMLeaLe091JVpflySRG75ycN8Wi8E0NE3SNCnAd/IaAK/V/tsJE9JkshQVJMDKw54URSVodE4iqJi1wFjOpOle0CL0FUVa/ezw24zPwuCwJ/+rVUWiyXTGExdSRTpDA3xrmXzzMj1eFUArb7w+otP465HnuOk+fWARq+IRqM0Nzfjcrnw+Xx4vd4DBmYPNoVgMrvoootYtmwZpaWlM5JDMF1/PtN2RFEI9leFYDIAGg6H9/hOELQKGcak2h8e7lgAK4oismzL03ONxNMMxZLUFnnyNBAVRaV3OIEkCvj1bNSe4Ti9w3GOqynQChaIgkktqCvy5B3T45Dzsv8NnVhFVU2FgJFEmr5IHEU1xqnS0h/VgYse6VXVPbipjSWevBtPlkROn6sB5aFYmkg8zdnHlFLocSCJAqs37kaWRE5u0JxrfzTJ42904rJL2jKQorKrL1/qw+uQWVSt3TzhaIpIPM1Fiypx2jTwumpdu1kkQRAEqqqqWLJkCX6/H0EQyGQyB5xm8HaiEBgPkPEA7IIFC/ZYXjLe1icCo9dff32e8zUSdI6C10PHBEGgoKCAqqoqSktL83wgTH8FYuz8Gs+fG20urNPKWZeVlY0r0bUv94TBmc9kFZw2iYGBATOaPDw8THt7Ow6Hg4aGhglLi1vvx2QyqevCam1b6QDWc2O32xkdHZ10edugGYwdmzXSKlnuPcGCWMdyJt2WRNiyAi0R7gcrL+W2FeeZAHEkluD5DbvwuhykMhlqywo4eb7G9T3ruDlmNDaV1vosCgJJXTWhwOcmEovrq3G5PmV1mbCF9Tkt17++tJmANxcR3tzaA8DoGBmxK89awm++/MEJzw/ka/9WlQRNGsJoPMmCei2huLCwkGw2S19fH83NzXR0dDA4OLhfK7dwaAHYmebA7os/n0k7YgDs/iZxTdSGqqqEQiG6u7uJxWJ53xuTxW63m1Wv9icCGwgEcLlcyDYboiQj6W/V6axCKBLH77JRoMuoGKBxIJpAFAVKfJr2KkDbwCjlARcVQT2hQp/0Nh3EGQ4knsqyoyeCTRTN7P/XWgeIpzLUFObe+v+0vgOf00aJvjS/oXOIbT0Rags95rn40+ud2CSRYq/TjBJv7hwyI6YGbUDVo64AfpeMrDth47uLFmscOkkU2NQ1TLHXwaXHVSEAzaEoz23ro1Hn+iqKyjNbesxkA4CA22a2aZRzXHFirXncx17v1PZVtSIJgUCAoqIiPB4Pil4q8j/Bmd0XO1g0Bdg7gJ1sm6uvvprVq1ebf69evZqrr77a/Lujo4NVq1YBmIl4Vlu1ahXXX3/9vg3gqM2YBYNBCgryq/vtjYo1kY2dO+MFE4w2nXYbojCxZOG+AolMJkM2k2bXzp2EQiFAU1bo6OjA6XTS0NAwadTO+hIXDAZNikEmoyUdGdtYH/61tbV7jQQavmjs+dB8q/E5N2ZrxTBxzOqSlQ8bsERjb3zPmWb/fG4nq75xHcfPqcEuy6z92RdYdoy2gvbtlZdz05XnAvDTz/4XT3zvBuw2mbQOGL1upxmNPaamxHJcSdeHnWN+d8+fXxhXyzelB1OyikJkNM5djzzHEy9uzDsfY836PJZEkR/fdBVZRcGpA3ZFUSguLqa+vp66ujqKiorIZrOEQiFaWlpob2/fZzB7KHFgZwLAWv0z7N2fH0w7YgDs/iZxjdeGqqr09vbS29tLIBDImxRj32xFUWThwoVUV1dPayJLkoQsy5SWljJ79myNLiAIxFPaG3ssmcEmiRR684FrTP896LZrKgCiQLMemWwozo+wDo6myCgKTrtsqgX8fWM3LrvErFKNs9Q9GGPb7gjL6ovMaOe/d2rVaa5YqhUWiKezNIWiLK4Ocky5X+PNNvWTUVSuOL7KjAqv3tBNVYFb15zVNGC39wybx1JVlSfe7EQWRdy6eHl4NGlW8DLsmHKfOeZEOktjiZfz5pciCgKvtYTpGopz3vwy87i/fbkVm5RrM5bKMBxL552LK0/QInuiIPBysybkragqNr08n9/vx+HQlg8NMDtZVvBEdqREYKeSFLA3p3nrrbcSiURMVYH29vY8bde1a9fmVXNZuXIl9913n6lAUFNTc1QL9hC08eae4RcPRBKX1ReMjIzQ09Nj/j2Zr9/fe8IAz+FwmK6uLjweD/X19RPSAcaT+zKoXwBOp9ME26qq0t/fb24/1cSiZDJpjusRnSuqvbTnIt+5Y+ePxWrWe9Rpl/OSX7MWgHTmcTmgaT3X6XR+JPnUhY0AXHvRcpYvnIVNFk0N2/m62sJjz7+JLIlce8FJnDgvt7IyEk/mqSUsqC/n2LoyGsuLUBQt0LJ2cwvf/d0/WLu5JdfPCc7Z2ADTky9uMAMb1mvkcDgoKiqivr6e+vp6ioqKzECVAWbD4fCUS2IfShzYfUni6ujoMP2tUQXRWvlwrH/emz8/mHZEcWAPRATWmkXa1dXF0NCQmaBg/D62D8Z3iqJitzvypFGMPln3EwQBRc29DRvi/4qqmpJUXqcNVVVx2SWduyqSySogQDar4pC1aK8kijT3jdBY6mNOuT/vHEUTaXqGEzSWek3g+mpzPyc1FvPOhRX6/gI9w3EqC9xUBLWxx1JZfE6RM+eWme1lFRW3XWZWia4aoI/xtNkl5thGUxm8DplLlmhJYOmswmgqw9K6QnObF3aEOGVWEZccV2Vych98qYUPnaotUWUVlbfawxR57FTrEd6sotI/kqKqQDKPu8zCd93SPcwxFX4+cEq9mUh2/wtNXHfGLHP/plCUUp+DoDuXeHFSg8YhzmZV+kaSVBW4kCQZj0crlJBMJs1sYKtjMP5NZAfLoR2MY0xlLFNxmitXrpzwtxUrVrBixYopb3/UDm07EMEEqz8fGhqis7MTp9NJcXExNpttUuB3IABsMpmku7sbn89HTU3NuHNbVTWFgnQ6PWmZa0EQzEStsX1rbm42JQD9fv+EiWHGs0QURZ59YyfvPGEefo/TBKs2WdKfI5OP38qhtY+J/FpVE4wXkXQmiyDofFsglclg06W8+oZGKCvIJe786fZPIIoCw1GtUMO8Wk254FdPvcKFJx/LR991Ul4EuL68MK+vn7nybM5ZMgeXw2b2c2ub9tJSUxo0x29EScfmL1jPvyAIPPPaNq44YwmhUGjCKLfdbqeoqIiioiJSqRTRaJSRkRH6+/vp7+/H4XCYnNnJro1xzJm0qQDYfYnA1tTUsHLlygl97qHsn4+oCCzs37KqwbsyhLGHhoYoKSkxwSsw7luZ8Zsoarp+huSTIAggiGaUVZIkJH0pxbiRDQ5pVlEQBUGTQBFy7QqCwNCozgnSl8EdNk3nry+ilT9sLPXljVtRVF7eFcLrtNFQ4kUAWvujxNNZTmosNts2KAcF7vzCBLKYiwCDBgDbw6Nj6AcZeobiJvhWVZVYMl8DVpbEPZLAzphbgiwKiIJglrk1wGsknuL1Nk3RoKbQgyQKPLVxN4OxJJUFLkRRIJ1ReLmp3wT6AHPKfAj6YlRc52UZ4DWWyvC3jd3MLfMRdNtRFJU12/sYjKXMxAdRFPC7NAenqCoJvQ2b3U4gECAYLMDlcpnzI5PJkEql9ikyeyDtUOLA7m2bo/b2sQMRTDDoWAMDA3R2duJ2u03N1XQ6PamfFwQhLxo3HVNV1by3/X7/XsGrVV1gvG3G9kcQBJPfraoqZWVliKJIKBSiqamJ5uZm+vv7x33OJBIJMtks37vuYvojo0TjSd75hZ9bBq79ZxsnUmyYOME9mkgkzN+sdAVREEzwCpCwPBfiidxye3f/kAk659WWUVbop07XhG3e3c9fXtpMQ0WhKVMGcOuKc/Ou42XLF+J22mjrHTC/29GpUTlqSgtQVPLKkFtXyYxrYeXC3vOZ95FOZ+jo6JiSb7Lrq3B1dXU0NDRQXFyMIAj09/fT2tpKa2srAwMDeZxkOLQ4sDMto3Wo2REzyn3NfrWa4XiN7Nry8nLKysqmPTGthHpJFPISrqxmnZB5JHw94jmsJ0QV+TTRblmWUFUNoA7HU5QFXCb3ydjn9bYBRFHgxMZiM8KazCjUF3txWjQEVVWlJ6JxlQxx7NFkhpea+nHYJFx2mayiRWwHokmT7wrw0Lp2XHaZ8qALFdjSPcz6tjBFXqcZif77Jq34gKGIEImnaQ6NmNXBVFXFIYt5S1h+l52ldRqnbiieZjSZ4YJFFWaC1+NvdGGTRbOCWGgkwZ9f79AAsai1FYnnP9zcdpmLdV5t91CMXX0jnD2vlIAuNfbwax0oiorXYQBYzPNuOHRJEnHp2rIGxcCYK+OB2YPh0MZm3s7kcWD/I7BH7e1jByICK8syg4OD7N69G5/PN+ESvtWsc3Xr1q309fVN65iqqtLT00MqlUIUxSmDV+u9Md79oiiKmQRsRAwNWb/CwkLq6+uZPXu2mQxngNmWlhYGBgbyeJmZdBqv20FjRRFel4PW3jBv7Owkncly7Q//YCbjtvfmJx0by/odoUG+cv9fGB7VfL+iA3Yrv9LKJ7VGbAFGLaA1akm2iidTeZWuVv/wU9SWagA2Mprgry9vQpYkXJaCCwvqy/POm6KrzPSGc0m6SZ2yUFtagCyJZLMKDocDu92er86jg9lkMpkHMIeGBvdpLtpsNgoLC6mtraWhoYGSkhIkSUvua2tro7W1lf7+fpLJ5JSlBvfX/lMc2EPZDstRTsS9gv1PbInFYqYwdnFx8bjbjCcwPZ5pN6UWYVV0p6CqqpaRqYMPI1prBblpXZc14LbnOURVVekIjyLqvxnfG6oBsWSapXVFpupAOqsdxzi20Y/mUJTwaJJyv9MEkH95sxOPQ+bEes3pdIZjbO+JcFJjMcVeDUC+sEN7IBjaq+mMwuvtYY6tDLC0rhBJFHhxZ4ihWIp3Lig3x/b7l9vwu2zUF2tv35u6hnhp1wB+lx1R7/9LTf0mJQLA58zxswwgefnxGjVBEgVebwtT4nNy6ZJqBAFaQlH+8mYXZQGneU5eburP27/c7zITwPqjSUaTGa5aVoOkVxv71Yst2GWREr/2wjA4mqI5FM1767fZbHi9XgoLC/F6vWZBibFg1rg2M22HAgf2KIB9e9pEc29fFGH2SFCSJIaHhwkGg9TW1k5rbu1LIpmqqnR3dzMwMGBWOBxvfJOB18naHrv8PLZvNpuNoqIiGhoamDVrFqWleuGAvj527dpFa2sr4XBYA0wW+kRNSZDd4Qg2WWLtllZTiuryrz/AVx/4KwDv/cYDzP7A7aiqysNr3uSXT71iargqOvDbW/8NGx7NRbZHLJ/jyZQZAU6lM1SVBM3Er0Q6w5o3d5JKZ8zEZNArhllodUYD0UQOgHr06my1ejTXWGkzCvqMB2YNGkAkEjEjsvvjJ202GwUFBdTU1NDY2EhpaSmSJBEOh2lrazNfTlKp1Iwm1s4UheBwtiNmlPsbgU2n0+bbVG1t7R7ZtVabsP61DlgN0962BXO52wBCAoIJ3AxpKCtv1hq1NcBY1+AogiCYINDgtEbiKVpCI9QXe/HoUUXjt2gig4AWhVRVGIgmeWlXiMYSL0G3HUkUWd+q3XyXLNEStWRJZCSRprrQzfwKjd9kZJieNa/MbD+rqNhkkRNqC/R+a1SD0+aUUOCxIwgC3UMaReCaU+rM/bZ0D7OwKsgps4tMwNsRHtW1Y7Xz9uDaFmySiE/nAfdFEuwejOVFaxdXa5nqmjxYhoYSL5cdX40oCKxvDbO5e5iTG4tM8P7wunZEUTCpG0G3DVV3hkarHzmtwVQr+OO6dgo8duqLPaiqVmlsc/ewWdMcNIqB3+8nEAzi8XjMh56x/GdEBWaCZnAwl62mus3bxWketcltPAWB6Zosy3i9XqqqqqY8x417bbp0MlXVCjIMDg5SXFxs8lXH22464NUKniYKhoxnBi/TALMlJSUoikJvby+7du1i586d5jPonCWz6QhpCgdlBT7zc2nQy3Z9+b280E8qkyGdyTK3uoQHbr2G+nItC99KVRibwGwds2HhSDT3eSS3XD+aSGEA0IFINC85yyaJjCZSvLS5xZS3AkzpLcOM0xmN5QNYl91GoW98UX4YH8wa0mXG+CYrLT4dk2WZYDCYB2aN+dbZ2UlrayuhUIhEInHAwexUVtzebr74iBnlvsq3gJbhadS2tuugZG823nEEcksuIUFZrwAAIABJREFUiqKagMtY5pdE0YzIGkDV+D2r5EdajXYSqQySKFBVkF8WNqsobOoawu+ym9n9RqQxHNUyVrUIrrb9i7v6KPLmihX0R5MoqsqJDUX5/FlVhTFco7SiAWVjPIqi0hGO5b0RZhWFvkgib7+g27bHTWwqCwgCmazCaXNKqCvyaMCzbRBFVfngqfV5ygKlfqdGV1BhXUuYbbuHTSCaHdM3gBPqC1lUFUAQBN5oGySazHDVibVmRPh/n9uJXZbwOrT+DcVShEdzJR0Bs/CCKAis2dHH7FIf8yv8CMDLzQO81TFknltB0PSAA4EAXq+XWCyGKIq43e4Z48webN7VVDiwbxenedQmtwMRgRVF0dRpnqopikL6/7P3ptFxnOe54FPV+75h6QbQ6MYOkiJFiqIokdpXS5YX2Z7YceJd2ZzEPs61597cOTeTe+bMcsYzvpOcez25cRInsRObcWxZGduSbdnaKVESRUrcQIIEGt0NNBq9d/VWXdv8qPq+rgYaKymSsvGeo0MI6Kr6aumvnu99n/d5BGFT1ThZlhGPx1EqldDT04NgMNiRArFR8ErK8YuLi1QuSy+vuNnrYjab0dXVheHhYQwPD6OrqwuSJOHkyZMQRQn/6RMPYUKzeQ36XJhJqfzRXp8LZzVd1aC/9T579LYb8NCBSVpu1wdxKlwuJaW/Fgcmh+jPxJgAALhqg/Zu6DXFAeB/+dyjYBgGO6OhtgzsmdlU2+fINeVqDfo7u8WMvq6WQP5691QPZmu1GjiOg9OpOn914sxeThAwSwT8u7u7YTKZUCgUEI/HMTs7i0wmg3q9fkXA7DYHdmX8ypzlVpu46vU6ZmdnIcsy7Hb7hm98J704/cPFsiTL2vqSEPBK5ErI3/RgligRkDAbDW1NYZKsIJ6rwMCy2NXvbTtujRdR40V47GZKI3gzrmZY79ApCgBAwGEBrzUrkf2eThbBMgzcNhMYqJzYp07Nw2kxwm1TwfCxmRxiuSoGAy1O7Hdei8FsNCDosUJRgJklDi9dWILdbKT7fnlazQYQseymKCNVqrcB3r2DXkDRuoC1lfontAYvjhdwar6IA0N+jPeqEl4/eXsB88UaQl57i4pwMUt5tuo+fXBoPuFLZXXC/oN7xuhz8ncvzSDgtMDvsECSFZxZKGGp3JpAAeCuMRX0EzWDW4cD2N3vUZvMTqdwPlWmBg8GowkjIyPYu3cvfD4fLBYLzcxeSTB7PTYObDdxbQewtQzs8u/C5WpqA+u/Cwh45TgOoVCIlu2XP8fE/nUjmVdFURCPx7G4uEgND8g4arXaCtC4mbBYLOju7sbIyAiGh4eRSi1AFEXcvXcUgApaSdNTj9eJTKkKSZIRCrjw6K07aVJEn5lcHuuNz25tUSEuJlsc4wJXbSU8RKnt2n/24dvwlY/e19bABYBWwJYHV2/Nv8GAG+HuVjW0VqttSKu1VCohnU7D6XRiYGBgVc7slQCz5Fy9Xi8GBgYwMjKC3t5eWCwWqiM8OzuLpaUl1Gq1LT/Xm+HArscX/1WJdyWAXYsDu5kHsVqtYnZW1ZcbHh6G0Wjc1PZrfZY+SCxLASjhoBLagEFrPNJPuCzDwGxsf/gIoK3yAs3G6rvwidGBzWyA3WKEgWVwKc1BlhXcHA207UuQZMxmOLAsozVqyTiVLGBmicPOPg8Ff0+dWoDDYsSDu/qgKApSxTouLnE4OBxANOCgnFhZUfCxAxF6Hs+fT2O4x4XbRrthYBm8PpvDhXQZh0a7aFb1H46qXNN+nx2KAswXanj+fFoT3WY0Z7H2VavLYsKuPjWTUBck1JsSHtnTh7DPrjV4qQYFtwyr51usNXHktTgYprW4sZoNtNGAXHOiVlBpCPjFuTR29XlUUwhZwS/OpZEq1mi218AwCHrU8qKkqA1j77khhAlNE/f7xxNIlxv0GhqMKl/W7fbAbre3cesuF8xeLQC7kUnz123Vvx1qXCkOrCiK4Lh2d713GsBKkoRYLIZKpYL+/n4EAq15Ug/AiSrNRsFrOp1GLpeD1+uF2+2GKIpoNBqrbrPVIFqmgjZ/SLKM//SJB3HfvnEAatf+UNAPBcDjj9yGv/sfPw5AzbJuBAB2ohMA7fS56USa/qzPmoqiRBVuGk0B1QaPr3zs/hXH8DjtHe/RSF83zeLuHR3AQE/L2KRQKLRxgjudC8dxWFhYgN1upxSUtTizlwtml8/FBoNqkNPf34+RkREEg0FYLBaUSiUkk0nMzMwgnU5vCsxutMr161YN+5U5y81mYDmOQywWg9FoxPDwMCwWy6a7Zzt9efTcIQLY9AoDtLNdWw3rZU30WVkSTUmmgNZqMlLt1KYoQVYUZLkGTAZWVSqAynMVJBljQTf0c60kKzg2k4XJwCIScEJRgEKVx6lkEbsHfBjpcYFlGVxYVMtCj+xpNUzxooyQ14axHo1/q439zoleMCCSXOq53q3xZAGAFyQcGApgMqiWAqcWyxAlGZ861OKa/uzMIvp9dtw+1gOWAU4mCvjlVBqDAQdVNPjx2wtgGIbKw5gMLKp8e6PUB/YNwMAyMLAMXpvNwWs34yM3q3I1yXwN//hKDF6bCUbNcezMfKmNr+y0mnD/DjUDk+V4xHJV3LejFz1uVYD8yOtxFGpNODS1AgZARRsDAd0f3h9GyGOFgWXwvTfiECVZXXwwoHqCbrcbNpvtssHs1czAbk+a27GZ2IyMVrPZxOzs7ArZq8vh0a4HYEVRRCwWQ61WQzgc7ugmRrriN5N5XVxcRKFQgN/vRzAYbOPDv1NBjiGJIgIuO/7Db94HUZLxxQ/diWP/7UsqPU2SUa/XUa/VLtshTX8+H7lnP/25zrfehU2xBQIlScZCpkD/9t1fvkl/dtos6HSLdg2FUKyoz8POSBCDPT56LwOBALq6uto4wXowW61WMT8/D6vVioGBgY5z0pUGs2tVqQwGA9xudxuYtdlsKJfLbWC2Wq2u+bxvdL7/dauGveuMDFaLzTRx6YWxo9EoFTneyqS52sOr54a23EBa3NZOQdxHJFmBKEmwmIxUR1Udn6qV6rWrMk4sw8Cnlb4NLINSvYmA0wJBaqcGnEoWMBl040A0QMczm61gqMtJ1QzIuEZ7XW3jlGUFuUoDIU+LXsE1BFQbArrdNtpwFs/VMNztbGtGy1V4BL02CtJHe5yU3E+u2UO7VCkVA8sgX+WxN+yj1/Tl6QyGup14ZHeInuN/f/4ifu+uUQScKk0iw/EoVHmMabQCANg36KO0jHyVx4Dfjk/eFgUATKVKiOfreFA7riwrePp0Co/s6aPn53da4NQa4iq8CJOBwUcPDKrXhWHw9y/P4JO3DSHoVrOxVV5EMl/DWNBFz/V/uHmQ3rcfnpjHB/f1a5l0hrryENAqSRKdLMl/5BlazTTheqIQbAPY7dDHRhMBPM8jFotBkiS4XK4V+9hqWXctACsIAmKxGJrNJgYHBzv2O+gBLFnArQdeU6kUSqUSAoEAuru7rzqA4DgO8Xgcbrcbg4ODEBV1UVwul5FKpSCKIhiG0SpCbjidzg2Zsaz188cfOEh/5psCFEV9R/GCiKYgwqzJMz597DTGwup8K+ruqd1igqwoYNF+rcI9fpQ1mS+nzYLfuPsmSLIMo6an3t3dje7ubvA8j3K5DI7jqGMmoILGUCi0oTK6fn4lc68kSW3NaySDS/7tdK02cr8Jr9vtdlPN2kqlgnK5jFKpBJZl4XQ64XK5YLfb2/a50fn+120u/pU5y43ynpYLY+sdOjaTOSCxfIW9/AFry76uoQkr6xq3DCxDye4kCymIMpqCDK9dtZQ1G1jKc10sqV92u8mg2qIa1CxjMl/FzBKHvYN+mI2qi1W5rmYNB7wqiCKAq8ILWCjWwDKgygkvTasUgaDHDjBAQ5Bw5FgMLqsJQa/aFXo8lsebc3lEA2qTmaIo+O5rczAaWAS9NkABFksN/PR0CkaWhcGgAvTT80UIokzPWQXt7dqmh8e6EXSrsljxnGqk8Ht3jdLP/MvrCfS6rRjXHMjeiOXx0sUMTAaWgvcKL7Y1eE2GPHhgp5olPhkvIJ6v4ZE9ffS+/N1LMzCwDGwab9ZqZFHXZVoB4NOHhzVqAoNvvxqDw2LEaK8LigKkSnW8NpNr4zG//8Y+el+PatJekqyAYQ2w2+1wOJ1wOBw0I7CRzOz1BGC3KQTboY+NgM96vY6ZmRnIsoyhoSFYLJa2v1+ukkGn7Um2t9lsIhKJdASv+gpaOp1elxNK5LdKpRK6urquCXit1+uIx+MwGAzo6uqCIAi0mczpdGJ0dBSRSARerxe1Wg3z8/O4cOECkskkyuXyuvdqrUw2Cb/HSTmtgiBS3VkwgNfZUhD4yJ030p/NRuOqPFi3w0YVZPq6PPQ9qh8L4QQPDw+3qVVIkoTZ2dk1aQadgmVZGI1GOg+bTKYNZWY3CmCXH8vlciEUCmFkZAR9fX1wOByoVCqYn5/HpUuXsLi4iEql0pbU2Aaw7fGuzMCulfFc7cuoKKrX8dLS0qr2gFsV4N5olkpR2oWhyXaKooCBWm4xGNQxGDV+LDEtCDitMOrHxgCJXBWRLidCXvXLbjKqAFaQZJydL+LGQb/ugSbgWN2cNFPxgoTnzqfx0A19cJiNUACcmS/CxLK4fayHZj6fOZvC/TtD+A2tO79QbSJb4bE/2rKJfXk6g30RHz52S4RmcX/09jwevbEf3a4gGAY4lyohma/jgV1BCni/dzyBx/b2o8upvsRqvIgTc3ncqvFoAaDbZVXdynTagR89oFIEREnGdLqCm6N+Ot6fn1mE32nG3rBPlTCTFbx0MYPbx7opaN8z4IWsTaDxfA1BjxWfvX2Yjv2//vIC/ujecZg1w4V8tYlKQ0Qk0Fod//atUe26MvjJqRQe2R1Cj0vNzJ5ZKGG+UMN9O1qC3TdH/PTz5xc5TARV4Gs0GqlFJpG30Xc+L8/MXk3x7G0KwXZ0itWevfXAZ7VaxdzcHAwGA6LR6ArwSuJKAlh9tjcajcLhcKzYhsxHLpcLjUYDhUIBhUIBJpOJZs70EluKotqNcxyH7u7uTUllXanQg9dIJEIbx/TBMAzsdjvsdjt6e3tRq9Vo5pLjOJr5c7vdcDgcG/4eN5tNmgD69COH6e/NZqNqew5VReAT7zlE5yurzshAfdet3C+pEsqyDFGSYTWb1kwu8TyPxcVF+jzJskx1YElm1mazwe12w+VydbxGy4NkZklfzFqZ2Y30Cax3LKfTCafTCVmWUavVUKlUaHaWZVlatdsIzlhPNeZXKZh1JokrK2R2hYJouulDEAScP38eoVCojZAPtPhJuVwOHo8HAwMDHW9wOp1GJpPBrl27Nv0ALJ+EZQ2ULt8PAVitL2lLmaDV3MW0fVZf0pdkBVVeAMswcGo6qfrs85uxPPYPtZ8/oLpLmY1qg5hBA+o/O7OIh24IaQ1hqlsWL8ro0YT8GYZBodqEz9HZA1qQZGrHSs9bAzyy0tK+JYCRawhU25VhGJxKFjEYcMBjM9FzPPJaHB+9ZZBmMJOFGl6azuA3D0ao0sDp+SImQ25YNBAuKwoaTQl2i8oRbooybGYDPc5P3l7AgSE/Ag4LVX791isxfOrQUEvBoalq5tq0yde47Lz+8WgMnzwUVUuLCnAhzUGSZewMeei9EUQZJo3y8VayiBsHvFpDF3BsJo9CtYmHbgiq91RRjRL8mmZuXvtZf72IliEBseRfRVFQr9eRTqcRCoU29dLZbMTjcbAsi4GBgVU/k06nUSqVEI1GV3z3rmH8qs3g1+Vc3ClDubS0hKWlpY7zaLlcRiKRgNlsRiQSaRP4Xz6HzszMoL+/f0vjOnfuHDweD/r6+tBoNBCLxaAoCqLRKAUD+iDgVQ9O9ECIaLpaLBZags9kMqhUKujt7YXf79/SOC8nNgJe1wpFUdrArCRJNDPo9XoRCATo/SXAvVarwW5XM6rVapUuBHieh9lsXnG/FUWBIAg0W0u2BdR3hSgrK5uWJYmW/3PlKgJu9RiCKEJYVvFsNpuYm5uDoiiIRCIrniGe5+k9JOeyWTCrD30ygczFS0tLNKu/Gs1gK0HuD8dxNBNLaCBOrWq3/FjT09NgGAb79u27ImO4AvGOzsPvygxsp1itiYuskovFIvx+P0Kh0JqZA7LNZgGs/ksHtDdn6fdHNF+JOoH+X/J3EnqgC6jk+BzHI+RrdW+SUvn5VAk7+70rwGuNFxHPVzERbGkqvjGbxZ6wDw/dEKLHEUQZbpsJTbE9s2dkGTSaEqxaSV2SFUylStjZ56HglWsImC/WMNbjpsoKsWwVg377Mr1WAQ6zkQK+HX3uFRnij94ySMd0dkE9zqBfzXieWSghVazj/p2t7O33jyfx3j0hWvKXFQWLpToiXQ56Dx7Z00fvwbGZHA4OB/DJQ0OQFVVn929enMEX7x/Xbd+Ay2aAx9aaDD9xW0S7l8CL0xncPdFDNXNfuLAEq9GA/ZqLmawoGArY6XlkOB63jQQoOP3FVBpQgLsneuj9y1V5eO0msAxDFwYGjfMFgL4EFEUBz/PI5XI0Q0BeDmtxZrca2xzY7VgrOmVbV5tHSe+BzWZDJBJpo291isvJwJLMWL1eRywWA8MwGBoa6mhS0Am8MgwDg8EAr9cLr9dL+aTlchmZTAaZjCpX5XK5VvB3r0ZcLngF1PN0OBxwOBwIBoOoVqsUzJZKJSQSCbhcLvj9/o7XbWlpCUNDQwBa15AAf0VR6PtQTzVoNpt00cKyLCCvNAXSv0tZBnTeZJfNQ4IgIB6PQ5bljuAVUBccFosFXV1dbWD2cjOzgApmi8UiarUanE7nhjmzGw39/anX60gkErDZbBTUkr8TQEsqyL9O8/CvzJl2ktEiOn/FYhHd3d1rgldg61qywNr2spSbo42NZlUZYmjQktrSH1tW1GyioihoNEWYjQb0elulBFlWsFCogasL2NnvbTumKMk4HsvCbjFirFedYOO5KnhRws1DXW2ZU1lWkNJ4tBaTOnFkuAaen1qEw2qE1WyALCt45uwislwDkyEPlUn5h6OzcFlNGO9V+WSn5ov42ekUhrocMGp0iH99Iw5BkhH02AAGqDRE/PDNJFhGVRaQFdUYIV9puZcoioIBn41OipKsYFefB/ftUPmrp+dLyFWb+MjNYZiNKq3gGy9c0lQW1BV7tsLjual0my7sjWGvmiVmGKoqQMBrsdbEt1+Zw4DfDo/NQu1oF0t1uj3LMrhtRF0k8IKMeL6GO8d7cHNUtdL9t5PzeHMuD4fFpN1zBbWmQCdhQZJx32Qv7p3sgYFl8NSpFN5OFDHc5aQZ5jfnCpDl9pe/0WhU1Qs0gW6GYTA4OAi73Q6DwUBf2FfaNGEbwG7HZqMTnYv0HjgcDkR1jbNrxeVSCARBwOzsLFiWXRO8EtoOGXun591oNMLv92NwcJDux2QygeO4tk74jdqMX05cCfC6PEhmr6+vD2NjYxgYGIDT6QTHcZifn6efI5loANRCdXms1b2/vHrKdEjQ6bf1uVqJCP3zIIoiEokERFFsuydrBQGyxBRiucPZZjmztVoN6XQadrsdoVAIJpOJAu8rbZpAzt3n82F4eBgDAwNwu92o1+tYXFzE2bNn8eUvfxnf+ta3cOHChSsy978bwvDnf/7na/19zT9eqyAdossjk8nAbrfT1VA8HkelUkEoFNoQub5er6NSqaCrq2tLL2NBENacTIixAaut8AknVq8JS76s5GcCNI0aX4hkKmtNEUulOvr9DphNrA4kKzi3UELQY0Ov26q5g7FoSjItURvYVpPT6WQRAZcFHpsZijbGJ44nsD8aQNiv6r2eWyhBkGTsHfRRndnXZ3MY8NmxN+zTxgvM5aqYDLox0qM6nxyP5WAzG7E/6gerSVw9eXIee8JejPW6wABIlxv44ZtJ3DHeA5tmefuzM4sAw6DHbaXX50Q8j6DbRs8/4DDDbGRhYFk0BAkmA4v9Gr+UYRj88MQ89kf8CPtVMHs+xeEX5xaxe8BLqRu5Cg+X1USvndVkwJ4B1cErma9iapHDrSNdVDrrhydUUwef3UxVIAAFNs2wocqL2D3gRdCjqjP86/EELEYGkYCTZt7fShTQqzuPsV4Xgh71PJ86lYLZxGJnqCWB9uO3FzDa21I3ICW+YDBIS3YkE6v3b1/O2yLXZbORz+dhNpvhdDpX/QzHcWg2m+jq6lqVz3gN4j9f6wFc4fjzaz2ATtHJVrvRaIDjOPj9frAsi0wmg3Q6DZfLhcHBwVW7w5eD2mw2C6/X2/Gz60U2m0Wj0YDJZOrYJAa0u2ttJFsmSRISiQQajQb6+vrQ19cHj8cDo9GIer2OcrmMfD5PnZf0TUBXKmq1GhKJxBUFr8uDYRhKlfD7/W3f/6mpKfT1qU2pHMdR3q8oijAajetmYEmQe61mDNvnJVEU2+cyXa8IWWwkEgk0m02Ew+E2WsJGw2g0wm63w+fzwe12w2g0otFooFQqIZ/Po1qtQpZlOq8uD3IfrFYrwuEwleIiyQSyDcEq+h4Gco03E81mExzHwev1wmQywWRSNcZ9Ph/sdjs4jsPXv/51TE1N4dlnn8X3vvc9lEol3HLLLRt6Bo8ePYqvfvWrePjhh9f93He/+10MDg7C4/HgyJEjSCaTGB0dXW2Td3QefldyYMnqZnmcOXMGfr8f3d3dmJubQ71eR39//wqdv9Uin89jYWEB4+PjbdysjUY+n0coFGr7HWneUsFhB5cXHWht+72sQIHSpmJQa4qwGg2o8gJcNnMbN1ZRVLOBbpcFRqpQoII7s5EFV2/CZTVTQDid5tAUJOwZ9FHO56uXMrh1pLttHLwowWI0rJAAEyQZBpZpWx3XNR6q/hxkRWnjk+ozetkKTxu3AGAmw4FlWES7HPR4//pGAvfv7IFLc4DhRQnfeW0Onz08TPeT4RpgGQYB3b701IsLixzGgy56rV+fzWOhVMf7buyjWU9CC9DfF0mWYTIYsFhqgGWBHpeV8pL/9sUZfGBvP3wOs6bLK2MuV8VIj7MjfeT7byZx13gXfHYLGEbly/78bBoP7w610UbIGJ87v4R9gz64rCqoF2UZ330tgU/cFqH7J9xlkm3Sr/T1nFn9xLlZmsGlS5fgdDrR29u76meSySRqtRomJyc7Nsdco9jmwF6FIM2G+iBUgdHRURQKBSrsr+8U7xTLQeb58+cRjUY3PaZyuUy52+Pj4x2zvVsBr/F4HI1GA/39/R0VDBqNBqUZEEtbh8NBS9SXC2avBnhdLQhInJ6extjYGADg+PHj2L9f1YItFotwu91gWRaCIEBRVFt2wtdfbX/6eZpEvV6nTa2A6uxlMhogCAJ4nkcikUCtVsPAwMAVp2+sxpklGt4mk4k2BJKmsbWqCQTM6/sXgM3TDCqVChYWFtbMNqdSKXz/+9/HG2+8gbNnz4JlWbz00ktr4p8zZ87g6NGjOHXqFDiOwze/+c01x/H000/ji1/8Iv3/xx9/HF/5ylfW2mSbA7vRYFmWSmispfO31vbA5ZWukskkbXhpdQSqf9N3KxKARjrkV+jDMoCBUZu66k0RTqsJVpMBLMPAYTW1bVPjRVR5AX0+OyRJPQYvSLCbWVR5AVaTFR67WhLnGgLOzpdwcKSLnmeu2kSv24oDQ11tAFOSZWTKPEJeNaOoaPzQpiRhwOeg4O/lixncOd5Deag1XsSxmRzumuihYDqZr8FuMsDvsoDRzr/eFCHLZnoekYCTdq9KiqoPSMwIAOCFCxncOd6NzxwehqKomrhHXo/jd+8a1car4I1YHj67CaMabUJWFAScauMYy6hA88BQu3KC22ainFZV53UWnz48BFZbRQecZmqcIGkLi8/dMUz38c2XZ/DbB6MY7lazFOW6gLeSBRwaaakofPimAfr5n59dxD0TPXh4t7rYKVSb+OHJeXzqUJSC08OjXTAwakb87fkidoTc+ORtEciKaqv7zZdn8cf3jrWoDQZ1widNE6Qhgzx3m9WZJbEZCsGvi33hdrRiLQ4skQEKBAJU2H8jIcsyKpXKluZhAp4ZRtVbXgu8kmd7PQAhiiLi8TiazeaaoMlqtcJqtaK7u7sNzFYqFTAMQ0HQVpouryV41Ue5XKY/k0wsACwsLFAepiAI684F63XL66uZ5WoDAY8DkiTRxXJfX987wj1ejTNLGhOtViuazSalcK1HhSH32WAwrACzyznXa83FG6midXV14cEHH8QHP/hB9PT0oFKprJu827VrF3bt2oUjR47g6aefXvOzJJ555hkAQDgcXueT73z8SpHWGIZBqVSCIAiIRqObAq9ke2DrAJZlWaTT6RUPm54jqJYUVPAp09+v7Nwk4NDAMlRUn4JdhkFDlCDJCkq1JuwWI/xOdVXW0Cz4CBD0asAVAE7G83BZTTg40tU2PrfViHpTBMO0msJ+eW4RBpZFn8a5zXIN/Mvrcwh5bRj0O8AyavPSpSUOd4x1U57pt1+NwW4x4s6JHigAEvkq/uW1OQwGHOjShP9/dHIeFzMc+rx2sBow/t7rcRhYhnJwa7yIuWy1TcP14JCfgsyaIMFjN1PwmirU8Xosj4PDAYz2qCX3fz2ewFSqTI0fJFnB2/Pt/NJDo13Y1ecBAJyZL6FcF/Dpw0OUk/x///Q8TAYWHpt6DxqChESu1gbsPnN4GCYjCwPL4IkT83DbTDikZbKThRr+5Y14Gw/3rvEesIxKqTizUILPYcanbotCUYBUsY6/+MUFGNnWi3Woy0EVLTJcAy6rCV+4bxyKApRqAr760ykKlBmGgdFkgs1mg8VqpSUt4jxjsVhouWsjnNnNANhfF+mW7dhYVCoV9PT0bBi8kudIFEUsLCxsmseXz+epxvdqWSo9gADW521vFLzqg4Dn3t5eqsHq8XhQrVaRTCYxPT2NhYWFDYN0Al6NRuM1Ba+NRqPt/D0eD/1Zrz5SKBRi9dfIAAAgAElEQVQoP5Y0dy0PwjPt9FTo+yAA4OK8alCQSqVQrVYRDAbbjv1OhZ4zOzIyQlUZCPhcWFhALpfblM6swWCA2Wym/xGgv95cvBHZRP0zHQgEEIlEtnLa64bH47kuwCvwLgWwnW5ivV6ntIKhoaEtlTIvF8CS7SuVSsffk58JYCVNXPrQa8MadCAXgNatrnpOmwwsWAZw21Q3E+IdbTKw4AUJDouRGh2cmVet/PZH2xUKKryAs/NFWEwG2MzqSvL580vIVXncM9mr8nVZBkcvZtDlsuKjt0QBAIvlBriGiDvHe1QuK8NgLqeS+/W6qKlSHWG/Ax87qP7u0hKHbIXHo3v7MdrjgoFl8KOT82iKMj58cxiSrIL7rz87DY/djMGAHYoCnJjL49mpRRgNLAX2iXy7j3TQa8X+iMqXK9YFcA0BH9kfxmRIdeg68nochSqPfWGVJ9sQJHz71ZhK7dDux3ivC2ZNBqveVK/nv3togt6Tv37+Etw2EyJdDsgKMJup4Nhsts204AM604Ljc3kM+Oz48E1hMFA1cL/x4owGTtV7HParWXNWcw0LeW34wr1qiS6Zr+EvnjkPp8VIM9myomr3AmqW2mM34SsPTQJQM79/8cwFmsUlXDbyQt8smNV3Fq8V20YG20FCkiTaoe/3+9HT07OpzCt55jaryZ3NZmkWMBqNdtxeD143QqMRBAFzc3OUa7kWD3y1YBiGNvmMjY0hHA7D5XKB4zgkEglMT09TYNbpfPXgdXBw8JqA10KhgOPHj2Nubm5VWTOXy0XBmNPppPe82WxiZmYGS0tLaDQabYsUYGOAzKxpsebzefT09GyYEnglw2Aw0Aa2vr4+2gC2tLREG8CuBJglHOJOczHZbrW4WtrgJFt75MgRfPWrX31Hj7Ve/EpQCIgwNqByazrp/G0kNmNH2ylo6V0jmtPVldLSRAWg4zC2a8HSzy3ThgVUez6LyYhul9qZbzK0NGOrDQF2sxEVXv2XAOOLaQ5jQTf2hP1tXFtBkvHsuUU8eEMfJkIeMADOL5bgd1hw10QPBS2VugCP3YzbRtsFuntcVlpSJ+fc47agKUpU009WFFiN7Y1p0S4nmhrQJr97dG8/Pd4S10CP24rP36MCuGpTxLmFMg4MBeg1+dmZFJwWE24dDlCgf+T1OD52SwQm7Xo7LAYUqk24rCZ6nI8eaFm7/uJcGvft6MXHb4lAUYB8lcffH53Fnzw4ScfONUSt8c1Cz/F37xqhY3/qTAoP3xBCJOCAAuDYTA4ZroGHd6sAVlEUTPa66EIkVaxjR8iDiaAbLMPgZLyIN+MFfPpwlIJyriHCZ29p0A747fji/RMAgES+hmfOpfGZw0Pas6Pg/CKHoS47VTxw20xtigo/PLmATx+KquMBQzmGkiTRiXYtmsG2+8t2bCZEUUQsFkOjoS6yN1vi1fc1dKImdApFaRnUuN1uDAwMUG7hcrckPXhd75luNpuIx+OQJImqfVxukE5/IlhPZKtKpRKKxSKMRmObYQKRTrqW4LVYLGJxcREOh2NdDjMJm80Gk8mEZrOJQqEAo9GIXC6HXC4Hs9lMz3G9a0qaufaND6Jaq6Orq+ua6EzLsoz5+Xk0Go22LHxXVxeazSalihCawXLO7HqxFs2A/Efm67W+E1djHt61axcOHTpEq9tHjhzBF77wBfzlX/7lO3bMteJdD2CJB7TJZLrslceVysASHmJLy46htAD9vtu1YNtNC5aT201GA0RZhpFMzoqCxWIN/X4HrCYDwABOiwmSrCBdrsPrMGMs6G5lGzRg/OxUGvftDOL+nSGVzqAoaAgyJoIeWjanq2dJVpuwiJ2erOD582ncuyOoZn5lBfF8FecXy3hgZ4g2iP3i3CL2DPjQ7bKAZRiIsowfHE/gNw5ENKUBBelyA4ulhqYKwFAHMT3gdZiN2DOglooUqJnHB3eFKOB9biqNPWEvPnZLhF6vr/3sPP7kwQn0uq2QFQVLZR6xbAUHhgK0zH77aBe9zgvFOvq8NgpesxyPJ0/O43N3DNNzfiOWR4/LgmiXg16be7WGLxkKZjJVHBwOtLmA1UUR793dT6kLdUFqk9LaO+jF3kE1Y/xGLI9koY4P7O2jn39tNoeDQwGaHQ777fiUBkZns1VMpcqUQ0ua0G4c8MDnUEGq127GJ29VS0i5SgOvzOTxqKaHC4alJVYi9bJc31CSJBQKBfo8NJvNVTmz2wD21zf0mbZYLAZBEBAKhZBKpS5Lymc5AO0UiqLavWaz2RUGNXoAvBXwOjc3B1mWMTg4uOWEyFpB1ERcLhfl/JZKJVp+J/rORqMR4XD4moDXUqmEVCoFh8NBFwabCVEU4fF44PF4IIoi5ZNms1lks1ns3r0bBqOprYlV/37UK1zIknhNnM4URbUKrlarCIVCKxZlZrMZXV1d7yiYFQQB1WqVfidWm4uffPJJvPjiizCbzatWoL1eL7785S9vml5JYjl14NChQ/izP/szlMvlLe/zcuJdDWAJYd9qtSIajSKRSFy2diBw+RlY8uA1Gg0KFPQTK9DKvi43MNDTC/QglgGovWyhyiPgtKLf74CiqF3+kqwADJCr8Oj12CgHltXA8VvxPEZ6XbhPMwFogWQGTVGEzWSgUl5z2SryVR77Bv0UdD/xZhLv29uPuyZ6VT4ngB+9NY/37xtQObGsKpvltJrw4K4QBXNPvJnAB/YN4CP7VXctSVbwnddi+NShYfRonNgXLyyhykt4cFeQAu0fnkji/Xv7YTEZ6fWazVYx1uui1+qO8R6Q5ux8tYlulxV/8uAEvc7/8nocHz0wiB6X6sB1fC6PmUwFH7oprGa8ZQWiJLe5n3W5LPjs7WqWM5mvYi5fx+HRLpo9/8HxJMaDLuwMubX7wlB+rIFlUOFFapPLMAyeOrUAm9mIOzULW1VbNofbx7ros3Bz1I+bIurn35jNQ1QU3KbxlGVFwZMn5/He3X2U3hAN2DHoU1+qsWwV+VoTd2uZc1lR8OSJeRwc8qPPp2Y4/A4LHtqlqggUak1Mpys4OKwZKxgMsGklOpKZZVkWlUoFuVwOTqcTHo+Hlnb1pT9SBtvmwP56B8/zmJ2dhSzLiOq6si8HwK5HIVAUBalUCvl8Hj6fD319fSuoWuSZ3Qx45Xke8XgciqI6O21EX/Ryg2VZmpWUJIlmK4GW3qnb7YbH49mSOs5WolwuY2FhAXa7fdPgtdN9MxqN8Pl88Pl8EASBSu85l52PrKNj5XI5iKJIwd/V1jYlCySO49DT07OupNt6YNZqtdL7vFEwSyoMzWYTfX19bda2y5txH330Uezfvx+BQADRLah3rBflchkHDhzAM888Q4Es4SInEgns2rXrih9zvXhXAliGYZDL5ZBKpWC32xGJRCh/pJMu4UbjclUIllMIgJY+3vJYrvu63JVLVtCWgZUVQJHV8nLAaaUZRDJRz2Y4jPa6qZSUCmplLJbqkGTgpmigDWjIsoK5XBXD3S547RbIigp+X76Ywfv3DiASUEHp67M57Ozz4EP7wxSULZbqCHpseN/efu26qVnF/dEA/VKpMlsGPHaTtmJjgHyFh99pwacOqdnNKi+i0hBxx3iLtvDsVBpjvS58aH+Ycku//uw0Pn/PGDVkmC/W8Oy5ND5+sMV1k2QFFV6A09KaGD6yX1ODAHApU8H+iB/7Bn1UpopriHjvnhAFlr+cSuOBna2Gkz6fnYLsfLUJWVHwof0D9Np/86VZHB4NYKRHHZcoy7i4VMbufh8FxA/vbrmA/fRMCoN+B24f61Jht6LgO6/F8fGDLWmsmyI+yBooP5cqw2kx4rF9A3Ri/5sXZ/DxWwapXFnYb0evW73nSxwPUVbw2E0D9Nr9w9EY7hzvwlCXyt/z2sy4od9Nr/8Sx2OkW+WsER3HYrGIhYUF2Gw29Pf3d6QZ6DUZL9cLfDvevVGr1TAzMwNA7T2w2WwbKneuF2tRCBSl5a64msIB4XVvBrw2Gg3E43EA2LA4/pUOnuep9jKxwdVnLTcLgrYSxLzAZrMhHA6vAK8b5XmuFiaTCX6/v+N7UZREzM7OUpMInudhtVqviTB/NptFoVCA3+/fNHXhSoBZRVGokoe+cW01mkFzmc3uOxGPP/54WxY2kUgAwDUBr8C7tImrWCwilUrB5XIhGo22SvWbJP4vj8ulEJAg2ohEUoQEyQgQMKKnFLRoBC06gX4cLAPqbAW0DBDmsqpEy6jmhEXK8Q1BwoVUGf0+B8J+Oz0/WVEwnS6DFyVEAk7aHX90egkBpwXv36uCPl5US943R/2wmNi262PQxk3+X5BkXFriVF1SbbKr8iIWijV6rrKsoNwQ2s7JbjbCZGzPTN890YOQR31xlGrqF5JwYg2sCjz7vXb85sEoGEZtjPrmSzMIemyUQvGLc2kk89W2BYDP3pLSEiQZd0/04FENvD47taTRIII00/pXz10EFFBOr8dqgiK3rr0oy/jM7UM0I/ztV2PIV3js7veBYdQmsX89nmhTPHhgZxATWtPbG7ECijUBHz8YoWDzaz+/AJZlKGVjtNsJv0PNUGQrPKq8iMfvGFYpIwC+9rPzaIoSbcDzO8wwG7SSriijyov41KEoIpqZw189dwmxXBU2LattMxvg1IAwkXJpNBpIJBIwmUwrMi/6BjCz2Qyj0YharXbZ35ftePfG4uIiWJbF8PAwLbVfbiWL7KPT9rIsI5FIoFgsrqtwQLiDGwGvxN2KYZirlnldHrVajdLhCHXB5/MhEolgdHQUPT0qbUnfOHSl3b8qlQqtanYCr4AKYDuBWDKOjSaROn9OTUJxHAdAfZcuLS1dNmjebOTzeUpNIdd9q0HALFEz6O7uhqIo9D7Ozs52bADL5XJ0kaZvXOvUAMayLL1mV/JaJRIJHDlyBADgdrtXZKGPHDmCxx9//Iodb7PxrjUyWFxcRCAQaJuYiMDxxMTElvbbbDZx4cIF9PX1we/3b2pbQhmYmZmBoiiUrO7xeGC321eUfhSto5wA1bUmWPJ3SZbBQO1gbzRFWM3Gtr8Dasn55FweNy1THFAUBclCDeVaE7sGWuYFZxdK2NnnWTGGpihjoVBDOOCgLlKnkgVYjCzGgx467m+/GsMnbhuiIGyJa+Anby3gc3eqDU+yrOCX59KwmQ24daSLbvePR2fxqcPDqssKgFJdwIsXMnjvnj7aoZ/heJgMDOV1AkCjKcGq6c1yDQEuq4mO/VSyCElRsDfso1nSv31xBvft6EXYb4dByxS/NpvDrcNdNEsqawsLA8tiKlVG2G+Hw2KkC4mv/nQKX3loko61XBcwk6lgT9i3gr/FMAyePp3CvZO9MLIMFO3z3zoawx/f39JtbQgSWEYFyPPFOnqJAYU27v/tx2fxH9+7k+6bZKt7PVZK4WB1z85/+dl5/OG9ozCyLFiWQa0pYr5Yx6iWXdUvihiGwX9//hLec0MQYZ+d0lYKhQItn0aj0XXLldVqldJ2BgYG1jQ7uAbxq5YSvi7n4kajAVEU27JIsizj7Nmz6O3tRXd39xpbrx7pdBqZTAY33HBD234TiQQ4jkMwGFyVE0nAAVFDcDqda5oJEPDKsiwikchVK9PrYzl4XSu7qs/oEbF9vWHCVvWYCXi1WCxrOqbpg3TPdzIr2EhYLFYYdEY3fLOJ06dOwWq1UitbvaEAyVpuxIp4q1EulzE/Pw+n09nGq77SQe4jx3G08ZFkZgl1wOPxIBQKrYsPUqkUSqUSTCYTJiYmNuyImEgk8PTTT+Opp57CmTNn8Pjjj2P37t14z3veA0AFqN/4xjeo9isAfOMb34Db7UY8HofX68Xv/M7vrHWId3QeflcCWEVROqbL5+fnwXEcJicnt7RfURQxNTWFUCi0qZIBSeeTfyuVCsrlMmq1GgD1oVwtxd7JiaQToJVkGaKkUJ1UWeM7thqsZEynOUyGVurjlWpNnEoUcPtELwVIGa6BbpcVTVG1YSXHEyUZPz+TwsN7+inInstWcClTwQM6busvzi5Sm1Uy3tdnczgw1LpuiXwVAIOw366jCCxifyRAm8BYlsF3js3hN7VMJAPgUobDc+eX8Dt3tgwKfnFuESGvDTtDHgqozywUcUOflwIwWePYmowsshwPp9UIq8lAj/2tV2J4bF8/zVhWeAH/9Ooc/uDuUXr+NV6EDIVmc/UAFQC++fIsPnO4BdhjuSpOzBXw2E0D9LO8IMFkUDmiU6kyJkPqhKRAbRL7m5dm8R8f2UH3ma3wMLEMPPaVixyGYfC1n5/HnzwwQfeR4Xjkqzwmg+6OE9tfv3AJv3vniJZdV7Viz6XKODgc6Ai4ATWTRjzPXS4XtShcbeJsNBq00aW/vx/BYLDj565hbAPYqxCdXBEVRcGZM2fQ3d295UUNKbXu2rWLUsPi8Tiq1eqaCQY9tYVwLUulEkRRBMO0zASI1NP1YBCwGfC6PJa7fwHrA/ZOQRajZrN5QwL9VyosVmvb+4+rVBCfm2sD0DzP03Mk73273U7P8UqOtVqtIh6Pw2azYXBw8Ko1pnYCs0TP1ePxrPlMZDIZZLNZGI1GTE5OXk923sA2gO0cZFWmj1QqhUKhgJ07d3bYYv2QJAnnzp1bc3W/PPRcFNLYQkIQhLbJZe/evQA0ojqzUk5rrWMsBxKyokCUFJiNLIo1Hg6LCSaNYqAHo6/NZHForMUxJaCx3hQhSjLsFhOlHfz4rSTetzdMx6coQCxbwUiPqyWt1RCosQLhlpGoN0VYTToPa1lBTRDbeKkq8FOW8XtbXGDCryUxlSpDgYIdIQ8FlN9+ZRaHRroQ6XKC0cb5ty9eoqYGgJqxLNaa6Pd1lmo5ejGLQ6Q5SwEylQb+/uUY/v3DO+jYY7kqDCww6G+pD+hB7UvTGdxOTRyAN+cKeD2Wx+/dOUJl0TIcj4DDBKPBgHpTpOAZUDnH33ktgT+6twXULy5VYDMzGPR31pz852NxfPzgoJY1Vrm9yUINd4330HHpn4HvH0/iw/tbnNilcgMvX8risX06wM3zmJmZQa1Wg81mo3qNBoOBvvD1YFYQBMRiMYiiiJ6enutG1HpZbAPYqxBr2XoTfupWIpvNYnFxETt2qN/HWCyGer2OgYGBVZtpVmvYUhTVzrRUKoHjOCpxaLPZUKlUYDKZrgvwGolEtgzGSAWQvG8IYCdglrhkvZNj2EpYLBYKVBVFQbVahSzLq46h0WhQNQMCZq9E9hlQM/Fzc3Mwm820r+ZqBxkDy7IwGo0U56zGmSV0SmKbfB1ZeZPYBrCdohOAXVxcRC6X2zKhmJS+enp6NsR7WQu8dhpvo9FAKBSi28pKC8gREEvAB2lMMmiNWstBblOU1L9BtaqVZIXqh0qygqmFEnYNeFeAX16QMF+oIdrtpBzc12ezmAh54LG1MoANQYLVZFgBSmu8iCWujrDfSTOhbyeLCPtt8NsttLv/yZNJ2sClKAqWOB4n4nk8sDNE5cIupDl0Oc0IOFtOWYl8laoaqPdEQVOSYTUZwAsSjAa2rRT+zNlF3DHeDbNBbXyTZQX/zzPn8ZX37KAZ5EtLFcTzNdwz2QJ5XEOA3awqLxAqAonFUh3PX8jgowcG6T6ev7AEm8mAWzQ5LkVROb1uqyrfli430Ou20nG9OpPD1GIZn7otSs/t7WQRO4JuSoEAWuB9qdzAsdk83ndjHz3myxezcJgN2BfxtTR8RRkmTY3gufNLuHuih4LTN+MFXFzi8Bs3D7bAqSjBrGXYX5zOUNc0llF5hkSsvVQqIRQKwev1UlkfYoNJwOz8/Dzi8TgmJiaoPM7IyMj12sB1XQ7qMuK6nItXA7Dnzp2Dx+NpsxvdTJAm3dHRUSSTSfA8j3A4vKpUz0bVBghIyuVytEJG9Fc9Hg8sFstVe55J1vNKA0cC2AmYJbbSeitbco6EPkFcvq4meAXUhi4CyGRFgaTJ+q0XiqK0ZWbJNlsFszzPU+B4rRYzRIqOYRhEo1Gqpbs8M2s2m/HSSy9RfrTZbMbIyMi6KgnXKLYBbKcgjVL6WF522mxspvS1WXkWIocSCATQ29vblqUkYK0TiNXzFyVZAS9IsGtle6DVMEGATYZrwGc3a3zKVoZUlGScWyhht8YPBQPEMhX0emxtNABAnUhimQqGl2VenzyRxG/dNqQ1o6n6peVGE/fvDNHz+N7rcTy8O6SNUeX4fuuVWcqTZRjgrXgRM9kKPqwpGwDAP706h4PDAQx3O8FqWdW/f3kWn9X0WAGVCpEs1jEZdLeV640GpiMQrTdFPHc+g4d3h+ixn51awlK5gd84MEiv6en5Im7o97QtECRJhsHAIl1uIMvx2NXvoef4z8fmEPJYcaeW9VRNBcqYDLrbbIFbYDaLKi/hvh299Pr+28kF3DHWRVUjFEXVwTUbDchXeSTyddwY9mpNYMCTJxfgshpx90QPzZhXGgLc2qLjQprDeK+LgtMXp7OYWizjc7cPU8BdqqvXh1w74i2ezWbR1dXVka+oB7N/+qd/itOnTwMAxsbG8Nhjj+G3fuu3rknDywZiG8BehdCLrOtjamqKcgi3Evl8HgsLCzCbzRAEAYODg6saI+g7sTcyF5Mue9INX6lUqHuivnfhneTCvlPgdXkQwE5AkCzLtLJisViwtLR0TS1qFUWhWUNJVqDI0qa76VfLPhMw63Q61wSzpKK0Ue7/OxGSJNGqVjQa7UgDIGB2enoan//85wGomdnDhw/jE5/4BG677barPeyNxDaA7RSdAGwmk0E6ncaOHTu2nP7fSOlrK+BV76e9PLurB48tGa12Vy4S+kysJCuQtGYsUevYJQ5dJBsrywouLpUxHvRAlGTahS/KChwWI3hBorxaSVZwOlmEy2pUy/NQG4WeeDOOx24apOPMV3lUeamN2zqXq7Y1BAFqiZwANEAFbqW6AJ/dTLc7v1hGyGOD29ZqxvrRyXncvysIi1HNqjIA/tsvp/HH94/T0vnbyQLOLZTx0VsiFEReSJcRDTjayvS8KMJiNKIpyijWmlQWCwB+enoRJgODeyZ7qELDd4/F8YF9/XBYiP6sgnKjCZ/dAlGSwYtyG+D/zmtzmAy6sUdnyPDkiSQ+uG+gbUFAmsTOLJRgMxkw3O2kC5O/ev4S3rs7RNUiJFlBqdaE32kBL0oo10V0uyz0mfinYzF0Oa24f0cvPfdEvobBgB0sw6BUF+DRXc/nzi9hOl3B5+4Yos+ULEtIp9NYXFyE1+td169eURS89dZbeOKJJ3Ds2DGk06o/+Ze+9CX8/u///prP/zWKbQB7FWI1AHvhwgUqw7SVIHM5yUatVhrdLHglDTpWq7WNZylJEgVA+t4Fj8cDl8t1RcHd1QKvy0Pv/sVxHH1/ejwe+Hw+WK3Wq1pNEQQByWSSVkxFWQYuUw5qs1QKSZIwNzdHF0nvhGnFeiHLMuLxOBqNxoZc35rNJp544gk899xzOHnyJM3MvvLKK5tuPr8KsQ1gO0UnAEvKTpOTk1ueFM6ePQufz0dL/ctjs+BVEATE43EIgoBwOEwn4k4PaSvTKtMMbLs2bIsrKssKSnUePodVBaYsAwZqRrbGi7BbjMhyDXS51K51BWrDF1cX4LQaUao14baZqRzXxXQZ6VIdd04G6ThOzOWxL7LyCyErCqq8CKfF2CpFNUVUeBEBp4Xa5ibzNfT77K2Mn6zq1Q51t8wImqKMakOATwd0CdBmWWZFVrUhSDi7UFb1UrXr8fz5NMp1kerSyoqC7xybwz2TvQh51ElZlGTMZKoY6XG2NTKRBcKxmRwiATuCHhs9///z6Sn80b2jsJoMYBkGjaaEc4tlClb1zwTDMHjqdAp7w14E3VaqMPG1n0/hTx5oNRUKkoyGIMFlNaFQa8JsYOGwGOk4/q+fnsfHDoQR8tqoasLFJY5a0AqS3MZ1/u5rcYS8Ntw+2kXB7LHZHG4dDtBnRT9GQM0uZ7MZJJNJOBwOhMPhdZ/jbDaLTCYDg8GAiYkJzMzM4PXXX8fDDz98vakPkNgGsFchVgOw09PTlEu42SCcbEmS1uS8Ep1XIlm43jNcKpWovnE4HF41yaHvXSDgwG63UzB7OdzIawVe9cHzPGKxGAC1s59I4ZlMJsqzfKepFMR2WJIk7Nu3T/2dJEORpSsmA9WJSqEHs3a7HclkEo1Go+3dfDWDaBpzHIf+/v513awI4OZ5Hj6fD8FgEC+88AIajQY+8IEPXI90rm0A2yk6AVhSdhofH99yGWBqagoulwv9/f0r/rYV8Do3NwdJkhAOh9tAK8u27DyXZ1UJiO1kNQu08yBJSLKMelOC02oCL0pQZAVmo0FVKjCwKFR5+BwWmhUlPMuGIOLtRBG3jXZTkEP4r01R0vRnWxnaYzMZHBrVbFQVBbkKj6feXsAnD7esV1+8sARBknHfziB9ev/2xRl89JZB2M1GMFAVAL51NIY/vG+cbjedLiOWreGBXcG2rOpwl7NNfYGrC7RjfzmgOz6XR7/XhqDHRkHhXzxzAb99awReuxkMA9SbEp48MY+P3TJIM8ZNUYYky7CZjWqp3WJsk6n6f5+dxh/cM0b5prkKj5+eWcTHD0Za90WSwTCqY9rUYhlDXQ5YjAY6jv/8/53G//y+liwQ1xCQ5XgMdbc3bJFj/vULl/DY3gH4nWawjArenz+foddn+ed/9NY8wn5Hm/3uD44n8ZGbW1kwSZJQLBYxNzcHi8WCSCSybqctefEzDIPx8XE4nZ0bzK6zuO5m8suM63IuXg3AXrp0CQaDAdFodFP7q9frtJwryzJGRkY6ZsU2C15Js4vdbl9V37RTLO+AJ6Vpj8ezZmNUp7hewOvc3BwAIBKJwGKxUN3VcrmMarUKoEWlIGD2SgYBYc1mE4ODg7RhWpIVyNLGOLCbDUVRUKvVaPZZr0Hr9/vR3d191a2wFUV1+ioUCujp6VlX+UhRFMTjcdRqNTgcDoyPj78b7AzP0gYAACAASURBVLu3AWynEARhhdA1sZYdGxvb8pfu/Pnz1PtZH5sFr81mE/F4HJIkrVqaMBgMHcdJMourSWxpjfNtIKZQ5eG1m9WmHaOBZmOJXBZXF+CymSgAM2jWr/s1hy79+XB1AUvlOqJaplSSFfzk7Xnsj/oRdNugaCoC3z8ex4f3D9LxxrIV8KKMiaCb7vPF6SXsDfva9Fp/ejqFh24I0QVIsSbg5YsZPHpjPwXqPz+zCEGS8fDuVib8my/N4OE9feh1WcGyDJqijOfPp3HfjiAFooIogxdVIE/ArT6eeDOJD+zrB7RrmOV4/M0Ll/A/vW8Xvb7pcgMNQUYk0Oq811+jn59dxAM7g/RazmQq+P4bCfz7R3bQzHmh2oTJyMJlNVHNXX385TMX8AUdJSJVqmN6kcOdEz0dwekTJ+bx4M5eamJQaYj459fm8Ht3jejULFoKD69cymHQb0e/r/XciZKECschFouBZVlEo9F1X6J6rdfh4eE2Qe3rPLYB7FWI1QAscecaHh5e8bfVolar0Wezp6cHCwsLGBoaasuMKUrLHhZoGXCsFYVCAYuLi3Re38pLv1NperXGqE5xPYDXZrOJubk5KIpCwevyEEWRglk9leJKuX8ROTTSlOdwOGCz2ei8KQjCFTVm6BRET7hWq9HjbuZeXqnI5XJYWlqC3+/fUM8N0Xq1WCyYmJi4JpzlLcQ2gO0UnQBsuVxGPB5fddW+kejE3doseNX7aYfD4TXHYrVadVadKxu6OtEHiEIBw6gArNdja2vqkmQFxSqPgKbzKiuAyaBO9CzD4FSygN0DvrZudADg6k08N5XG+/a1mqveiOXQ77Ojz9vOd40EHCsavwCgKchtXfaiJEOSW/q1gKaTamTbStwE5DVFGYIkL9OXzWOs1wmv3Uyvw9+9NIOPH4xQgFqoNfE3L1zCf3hkJx1PqlhHrspjV1+r5M+LEkya2H8iX6O8U0DVV33iRBK/e+cIzVC/NptDpSHiXo1vql4nAQ6rESzDYKFYR5/XRsd6Ic3hqdMpfFGXWVYtYQ2IdrUyl/r7+W8n5/H+vf0UEJ9ZKOHYbB6fPTzUUVv2+FwBNw54tAyxeu7/9ZfT+PP3t7K7vKA6qRE+sCjJqNeq1LN+tReYPt4FWq9rxTaAvQqhKJ01uUl5eGRkZEP7qVQqtBs+Go1CEATMzs4iGo3SjL8evG4080pAgtPpbLNFvpzQZ/PK5TJtjCIgjwAyEtdKY1Uf+kaljdrkdqJSXI6ZgB44DgwM0KY8/Tuw2Wy+owBWUVSTi3w+j+7ubgQCgRVNblcDzJKqFqn2rneM61zrda3YBrCdohOA5TgOc3NzGB4eXpcIvVpcvHiRrpKBrYFXUqLZ6ERBbDkBrOC5ElkqfXMUsFKxgGwLBSjVm/DYzWgIagMTAUFTqRImQx5Issox1WcXnzyRxAdvCutknepwWEyaI1V7Jjhf4eGyGmEytqgFL15Ywt2TrVVkhmvgxQsZfPCmAS2brOBkogBFVrAv4qeZ3Wen0rh/ZwsYNUUZ5xaKuGHAp+PJSmhKCrU9XR6nkkXsHmjx5ApVHj88OY/PaE5figK8cGEJGY7Hh/eH6bFj2QoGA462LC01eig3EMtVcWAoQIH+D95MQpaBD+0f2JD6wGy2gnOpMh7Z3Ucz50+fScFlNeLwSDfdhnCWAeD0fAk3aIoHDAO8OpPD0Ys5fOmBcUppyFWbcFuNMBsNyFZ4dOmUDLiGgK/9fBp//v6WlJwkSWg0GojFYuB5fkONAnqt1+7ubgwODq75+eswtgHsVYjVACzJso2Nja27D47jEI/HYTabqXxQrVbDzMwMIpEIXC7XljKvhLe9UZCwlejUGKXnkkqSdF2AV0Jl22qjUif3L2Im4Ha71+UFK4qCZDKJSqWCvr4+eDwtwx19JZLn+Q1b0W4lyDPh8/na1ICA9ntZqVTaFBuWa2FfTpAFDUmUrbeoIvQXhmEwMTFxPWq9rhXbALZTdAKw1Wp1xap9s6Hnbm0WvDYaDeqnPTg4uKlVkslsoYBweUl/eSyX1wJUEJnMVxHpctImMAIcF0s19LptasOWpIJRIrT/i7Mp3Lsj1N7YpIGtfJWH32Ghuq3TSxyyXB23jfRQMPVPr8bwyJ4+qiFrYBn8/cuX8OnDIxT4nUuVkcjX8NANLbmtHxxPYDLkxo6QmzY8/ZefTeFLD07STORiuY7vHovj3z00Sc9xOs2hVGviZp0e61K5gYDT0lamJ6C7UGsiy/EY621Jgj15Yh4GFnjvnj5q7vCDN5O4f2cvvBq3VpJVrq3XYYYoy6jxUptSwg/eTMJrM+KeyV6qYPDEm0k8uqePZpv10lhLXAOluoCxHhe9b//4SgzdTjMe3BWigDiZryGsqQksb2B7dSaHN+cK+Pw9LeOD6SUOQbeVjrtTEK3HSqXSlvlYLfSNAte51uta8a4b8DpxXc7FqwHYRCKBer2O8fHxNbcvlUrUvlRPaanX67h06RLVft0MeFUUBdlsFtlsFm63G319fVfl+SUujKVSiXJJARWgrVeJe6eCNBGLonjFuuwJL7hUKq1w/+okWaUoChYWFlAulxEMBjvSkMiC+p0EsIRKspFngsgHchxHFyZrZdk3GqSqRSoN6wH/SqWCRCIBAO82CheJbQDbKURRXPGgk1X74ODgut18qwXhbg0NDW0KvF4JP219VkwPYjsBWklWIIgSrGZVUWC5DakkKxAlGRVeQMBpBS+IMBkMNKN7Ip7HZMhD5aJa28k4ES/gZo0bKytAMl/DiUQeH9zXytAevZjB7gFvG7c1lq20lckBoKKpFRDgWq4LsFsMMOqyxzOZCqJdjjZKAXHKIp9J5Gt4I5bHYzcNQFZUy9kfnlCtTz+wt5+qKfz47QXcu6OHun9JsowljkfQbW0D/WS/z59fQiTgQLTLQf/2v//4HB6/Y0hVVGAZNEUJb8WLuCnq76hg0GkfX/3pFH7njmEKLEVJRpproN+r3uNaU4Td3KJJfOtoDAN+G+4a76Hn8tJ0BneOd7dnCbTrfypZxGy2ivdryguSrOCFCxn0e60YD7rp+BqNOlKpFIrFInp7e9eVWVEUBYlEAtVqFXa7HRMTE++GRoFOsQ1gr0KsBmA3YutdKBQwPz8Pu92+wvmI53lMT0/TBRdJVmwEvGYyGeRyuQ35yL9TQeS6CMcSuLzy+1ZCFMU2iaitViVXi41IVjEMg8XFRRSLxTUblcjYGo3GisTUlQiO4zaluqKPTsYuRqORZmY3CmZJVQsArTSsFXoKVzgc3pC50nUY2wC2U3QCsI1GAxcvXkQ4HG4rUWwmCHcrEolsGLzq/bQHBwe3rIDAMEzbCnm1TKwkywAYCqZkTSaLlPkVRcGlJQ6jvW5IkgyGbWVjE/kqPDYTfA5LG11BkhUcvbiEW4e7KGAxsAxen83hwFBr0iG6sQQg67mt6XIdAYelzRHsVLKAPeGWk1SVFzGzxGFnf4uXmq/wYBnA62hlrEu1Jlw2U0eebKkuQJYV+BwtPdmfnEphtMdJxfwNLIP/4yfn8Pt3D8NlVbv4a00JT59O4bF9A62mL42jazUZkK824bYZ28D115+dxicPRWEzqS+ccl3APx2bw+fvGaVja4oyZEXdR6pUh99hblMf+F9/fAZ/eM8YNRGo8SLemMvj8Gh3x4atf3trHtGAA7v7W8/wPxyN4bdubXF+FUWBKMswGQyYzVZRqP3/7L15fFvlnS7+nHO0WIslWba8W3YcL9kgJBAIIeyhBEqhQDstpbSTTmlph6G/ucPcmc6HthQ6dFq6zO1wu9ELHaDthGlpSwdq1gKFBAiQBZzNcWJb3mTti6Wjs/7+kN5XR45sy4tkm+r5ByLbR0fbq+d832cRsNldlXOcZDIJn88Hn8+H6urqWRdAVVXpl43BYMCaNWtWilEgH8oEtkTI14o4OjqKSCRCq2CngkQeWiyWnDxWAkEQcPz4cdTX18Nms6V3lGa5kNK6ugvJNi4WpmpeVVVFJBLJ2X5frPrT6aDNHi9FRFS+yCpShyoIwqxGpWISWPJ6FJq6MhPIlJ0kNhAySy5MpsvS1ebNtra2ziot1Eq46urq5l0IsgxQJrD5kI/Akqv2pqameY/aBwYGwPM8mpqaCrqy0rpL3W73gr/wGYaBwWAEpyEq2nMg/56qi5XktGyAyAi0SDd4SfDFUlRioI3t2nfKD1elEatrbfS4Y+EkGhwmCJIMPZcl8UlBQu9wBJvbnHRSuO9UAElBxsXdWWnB/3u1H9dvaqETSF6U8dNXTuDvr1hDJQoHhkLYNxDAbZekdXKKouLJg8NwWSuwdXU21/TJ/SO4/uzsB1iQFJz0xdFZV5kjoVDVdGSYN8qjxmrMmbT+4vUB3HReK8jb3RtJ4qevnMTd122gz+tELIVYUsTqWmve6ffLxyZwcXctPf/xSBI/frkf9374DHpu/ngKCUGC25n/C+Pne07h41vcOeaz3W8O4QsaQizJCpCJ4zowFEKz04waq5ES83996jD+7rIO2sIlyQqivAin5gKA53mEQiGMjo4WvI2qzXpds2bNcm3YKhRlAlsizLXWm5QUVFZWTqsBJATWbrfD5XLNuq5qL77y6RtLhXg8juHh4Wk1r6lUipJZURRnDNmfL/I5/UsJYnLzer30vcGyLCV5+bSkhMAmk8nT4jEXAu2W/WKnP2jJLGlyy5elq42/amlpmVXeqJVwORwOtLe3r0QJF0GZwOZDPgIriiKOHTuGxsbGeTVSKIpCJ1YAYDQaYbfbp40PIYvVYpFXApZlc4KkCYmaSl61U1CS3ar9ffL/vSNpU5S2WlZWVJyciCEhSNjodmaPnZnKkvgtAl6U8fi+QXxqW3u6ihbACV8M4xEeF3XVZksFjnqxuc2ZIy2YOsVNCjKGgpPors/KPPYPhuAw67HKZaXHenzfELa2V6PFaYaipifM33zqML78wXVUJ+uNJvHf+zy4Y0cXfY4mYinwggy3JgZLlBXo2PT2Y583hs66rA50IsrjiXeGcdslHTnpAyOhBK7f3JJbxWrUgeNYBCcFOC3ZSXtgMoVfvTGE2y/rpMfoHY1gJJTEBzS5rdrz2NPvx7bVNTlRWj9+qR/fuD5LiCdTEkRZgcNsQCQpwmLgMjXB6ffA3U/24u8u68hpPBsZGYEkSfD7/TCbzXC73QWHvK+wrNeZsGJX/GmwbNfifJncXq8XPp8vp9abOMB9Ph/sdjuam5vzvi9VVaVB96lUipI8u90Oi8VyGsnTRgxVV1fD5XItS/KqBdl+J2SWTCwrKytht9vnbRgi5JWE8y/V55ikP5Aps1ZLSrbf7XY7nViSYdFiamAFQcDAwAAYhiloy34hmC5Lt7KyEjzPY3Jy8jTzWj5oJVwrKOt1JpQJbD7kI7CyLOPIkSOor6+n4ciFQhuMrSgKYrEYIpHItE0spE+7WO7SqRmxU6tlZUWBqiJnux4aGYGsKDgyGsGG5txJtKqqiCZF9HmjOGdVDaZOeEdCCciqiiaHGQwDsAyD37w1hBvPybrQZUVFcDIFV2VFDikG8sdmRRICrBU6em6KqsIfS7vnWc0ENSFIOaYlAKdNhLV4/vA4dqyrp479ExMxPNvrxd9e1kmP+dIxL+K8hGs2NqbjyBQVA4E4WqosOWUQZLrpj6dwyj+JLW1OetsT7wwjkhTw6fNX0QuHA54Q1tbbaHoAec45lkVoUsBBTxiXrKmlcWfP9Y5jMJjA32xvp4TYH0+hyqyHjuPgi6Xgqsy+3r4Yjx+/1I+vZPJpFVXFeIRHOCFgbYNt2i+3QCCAoaEhug1HviiIHi0ftFmvq1atWo51hPNBmcCWCDPVeq9btw5sRpJDprIOh2PaVACtcZYceyrJs9lssNvtVG5FDEI1NTWoqalZ9uR1KlRVPS3KiRiGtCRvNpBK0mQyWZBZs1ggrvmp6Q/5tKRkYul0OmG1Wun37UJBkhcKjQxcTOTL0uU4DlVVVTMWQ6zgrNeZUCaw+SDL8ml5cYqi4PDhw6itrS1Y8DxbPAtZQInjkmEYGI1G8DwPo9FY1GgUlmWhNxhytK3aaQaZSgJpopeSZJgMOgTiPMwGHc0A1T7WPSd8uKDz9OdmJJTA8fEILl2bTQp4o9+Hc1bV5Og0yX2NhifR6LBQAhqaFLC334edGxopyXv9pB+ipODCrlqaZPDY6wO46owGOC1GsAyQFGU89OeTuD2Tm6qoKkbDSew7GcC1m9JxVYqqYiSUhFHHoNaWJbHaPNaUJKd1pxmiPxxMIJwU05FUmdt+9foAbCYDrjqzgdbdPr7Pgx1r61CTIY+yoiIYT8FlS0+fp5qt/vjuKIw6DpetrUu/dwA8tncA2ztdaM80aimKimhSgMNihCQr8McF1Nsr6DF+t38EwckUdl2wihrLDg2HsdplobIA7esd40X8/sAoPrm1lU5r3xuNoM8bw/WZ54iYaXiex6lTp6CqKiwWCyYnJ2fMNiS1kis063UmlAlsiTBbrTfHcRgdHUUoFEJ1dfW02lQteZ2qeSUkLxKJ5EzyWJaFIAhwuVxzHlosFhZCXqdiJpJnt9unJT/ajNVCKkmLBWJem600opjtX9qmr9bW1iVJfwDSzaBer5fefzKZBJDe2SWPU+uX0Wa9dnd3r3QJF0GZwOZDPgKrqip6e3vhcrkK6mefSzA22fLx+Xz0Azd1GlCMK3+WZWEwGjVmq9yCA3JbOrKKyZnIavWh7wz4saXdddrxw5Mp7Dnhx9UbmyjRGw0n0OgwZ6apCowZA5OsqHj5qBcXdLro/SQECb96YwCfu7iTxma9NxJBOCHgQo204OlDI9jkdqLBka14/e99Q7jh7Ow2Ii/K+PVbHnxq2yp6rFf7fOjzxvGZC9ONPmnSOYQGuwnbO11UJ/vc4THs3NBIH5ekKAhNClQGQQxghBS+cMSLzlor3NXZ5IBv/OEw/np7GxrsJnAsg5Qo46WjE9ihkQAomfcMx7J4byQCV6URdbYKeoz7njqMD29qQne9DRzLQJIVHBgOYVOLM69h66Vj6drdKzJZuOk4rhGsb7JhjUZiQcxzSVHG/sEQtnXU0OkuyzAQBIGSUUmS0NbWBqMxXRmsjYMh053Dhw/D7/dj7dq1tEpxBWa9zoQygS0RZqv19nq9iEQicLlcqK2tnZW8zmacVRQF0WgUExMTdPAwm9yrWFhM8joV+UhevsepKAqGh4cL3qYuFojT32Qywe12F7z1nW9iOd/XU0vkl1JCQYi81WqlUhlRFOnjJGSWZVk8++yzaG5uRnt7O528rrCs15lQJrD5kI/AAkBvby+cTicaGhry/FUW8wnGJlsjJpMJTqfzNF0P+cAt9pXTVDmBtp0LGeJNXsepkUtD/jgcZkOOwx9IazFfOjKOKzSkjyAlyhiLJNDitFKC+MdDI+isr0R3vZ2StVeOeXFRd/ZCQVFUJEQZ1jzlB4qqQlHUnKxWUVbAMQzYDNHTcWw6p5ZjISkK4rwEhzmbNNA7EkGVxYBGh4kS41+8PoAtbU501lXSPNl7/vAe/mnnWuh1LBgAsZSE53vHcV1mWgmkjWCyosBk0CHGi7AYdNSAxjAMHvrzSXz0nBYqEYjyIv7j+eP4qqbtKinIiPECam0maqLTGsd+9KcT2LmhHu5qCxgmTaJ//top3HpRNldVVlQoajpN4Lg3BgPH5sRx/eCF49jsrsL5GlPbRJRHnb2CXtSIokjb34j2Ld8CqA3q/sIXvkDzBbu7u3H99dfj5ptvnneCxjJEmcCWCDPVepNdgLq6Orhcp19AA3Mvi9ESNtJhP5Pcq1goJnmdCkmSaPbq1FasWCyGRCKBhoYGOByOWY5UHGid/vlSJQoFIXn5Huds8WPasoSlnEInEgkMDQ3NmHpAWs727duHL3/5ywDSjWQXXXQRbrnlFpx77rmlPu1ioUxg82G6Du4jR47AbrejsfF0YkYwH/I6XZ+2Vi8701XyQsEwDPSG3LB+YhbKTloVCFKalPGihFhShMuWJXvp31Gx98QEtnedPqHmRRlvnPDh4rX1dHt8/2AQDQ5TTpVsUpBhMnBIChKMei7rnlcUHB6OYEOLg94WnExhb78fOzc0UgLWOxrBJC/h3PZqeturfRPY5HbCZko/X5KsYP9gEGe3VedEXoUnBbq9T0Amun3eGNpd1hwS+fPXTuGvL1hFTV+DgUn8/NWTuOf6M9N/m5En+OI8zmquovclyQrYTPzYgaEQzsw8JiWjIf7+s8fx9Q9vyLyXgNFwEkfHorhUUzlL8moZhsGLR704p9WJygodFFWFpKj4+h96cV/mPIB0zW1oUkC93ZQh2CpMBo4+lu89dwxnt1ZheyZ+S1FVyJIEQRBo7mYhC7eqqti3bx+efvppvPHGGxgfHwcAfPWrX8XNN98849+uIJQJbImQj8CSjFcAaGhomDb/cz7klUzYphK2fHKvmcxfC0EpyetUkFasSCRCM3iNRiOqq6tRWVlZctMPyUBfbKf/TO1flZWVOfej1Y8WknddLJAmTpZl0aYp5pgOPM/jiSeewJ///GccOHAAPM+DZVkcPHjw/TJMKBPYfJiOwB49epSO7fOBmLQKDcYGslqW2XQ9+a6SF2saIMsyxsfH0d6+Om/EliQrSAoSKk0GCKIMg57LmYIqioqjYxHUVFagdgoBlGQFf8pMY8nW/USMh0mvg82khygrOfmjnmACsqLAXW2lROqRPSdx49lumDO620RKwiN7TuFvL++iW90HhkIYDEzi+s0tlHTufnMQ7TVWbGmvplPH7/YcwacvaEeVRQ+AQTwl4gfP9+ErH1pPCeNQcBIvH5/AJ7euyjZyxVJgocKl0ckKkgJDxqx1yhfHKld2S8kTTOD5I17sumAV1Za+dHQCpwJx7LqgnT62sXAS1RYDKgynL0aJlIT/OTSKv9ripiT57cEQXjk+gf91RTclxLGkCIOOgVGvw0SUR01lWhZCXsOv/PZd3Hv9GZT0hhMiDg6HcGFnbV7pAXlPpFIpeL1eBIPBGYPCtX+vzXrt7u7G8ePH8c477+Daa69d8ARnz549eO211/Dxj38cLS0t2L17N+x2O3bu3Lmg484DZQJbIkwlsLIs4+TJk0ilUjPKueZKXkktazKZnHGrfDqH/2LJvZaSvBJop41msxmCINAiAaJ1n8m4uVggMVUcx6G1tbVo8g3S/hWNRilp18aPBQIBBAIB1NTUTDvpLzZIcoaiKGhra5uVgGqzXsna/fLLL4NlWVx55ZULPp9lshaXCWw+TEdgjx8/TjuGp2Imk8B0IHEgVqsVTU1NBV/dLuY0QBuN4na74XRWUxIL4LTteiBNWGVVhZ5jEU4ICMR4rK6zTZnGKnij348t7entaTItTGWyX/0xHtXWCjolPTIaxvHxGG44x02P89ShEWxtr0G11UjJ1SFPuryAQFIUTKYk2E0GSlwjSQFmgw56TSTUUGASzVXmnHzbqRFcCUFCz7vjuOHs5qxO9oQfBz1h3K5JH3j63VHoWQZXaGpaj41H0VVXmTPFJtKFcELARCyFrrpK+th+/dYQEqKCW7a2UrPV80fG0eGyYnVtZc45mQ06SLKCfl8c3fXZ5/lPR71481QQ/3TVWnpu/b4YKnQc3NX5dU6P7h3ALee3UULsCSXwm7eG8fdXdGUnxJnJK3l/Fpp9Weys156eHnzpS1+i//7sZz+Lf/zHf1zU+ygQZQJbImgJLPkSJxfwbW35a73nQ17JGjiX7eHpzF92u31GU9R0WC7klSQvEOJDigTI49SS9umyVxcKbUzVfNsn5wpVVXOydLUyQrPZjObm5qLKRqaDoig0u7UQ41gpsl6XyVpcJrD5MB2B7evrg8FgQGtra87t8yGvxBW4kD7thU4DtAu3NhpFbzBAn1k8yVRSq+HkRRk6loE/xqPeYc4hroqi4shYGDaTHi1Oa85UL/2zCNY3OahEwRfjcWAohJ1nZGUZp3xxuKstGXKYS6D9cR5VZiOdHEqKgnc9YWx05zZyHRoO4VxNyoE/lkKMF9BabaVEzR/jYTfpoddlFyVelFCRMZbRqtrM4zs8EgHHMTkk8sFXTqDdVYlL19TST9NPX+nH9ZubUasxeQ0GElhVY6F1u9r//vn4BGwmAza2OOht33v2GLa2V+M8jRTi+HgU3fU2msSglTO8MxjEQCCBGzY309esp3cMLMPginVZo1hCkGDSc2AYBm+eCuDcVdXp7N2MYYtMXolRYGpczXQoRdZrT08PDa/PdxFZQpQJbIlAIg3JREkQBNTW1sLr9aK1tfW0OKe5kldtq1RTU9O846EWKvfSktep1belgnarfLrkhelI+2xtUXMBea1VVS15TBWBqqrw+/3w+/00A5xl2dOqbEtxHiS7tZD4slJlvS6TtbhMYPNhOgLb398PjuPQ1tZGb9MmDQAoqJKwGH3ac50GaBtV8i3cRqORLqKkiSqSEFBdWZE1RWnSCADAF01iLJLEmS3O00jt4dEIuuorM2Q07bQ/OBTCRndVzu8Cab2sJziJdlclJW9vnPRDVVVsXe2ixO1Hf+rDx85tpY1ciZSEn7zchzt3ZssIDnpC+HOfD1/a0U3P5fcHhiFKCj5yjpsSyV+9OYit7dVYVWOlxq8/HZ3AjnV1OVKKcEJATYaYErJJSOQrxyewrtGOGqsxWwbw+/fwuYtXo85WQStnf/XGIP7mwvac4/KiDGuFHqGEAKOOzYnX+vfnj2N7Rw02u9M6WkVV8dieAXzy/DZKxhVVhSiliydGw0lEkyLWNGTbzx569SQ4lsUnt7bS2DFPKIFGewUl8LKiIMXz1ChQUVFRkOO3VFmvPT092LZt25IZKDQoE9gSQZIkJJNJuh1KTDz9/f2n1XovhLw2Nzcv2kXXXOVehLwu1KS0EGjlP4VulZNYrkgkQtuiSFyV3W6f19RU3mC4ugAAIABJREFUkiQMDg7S13qpYqpI6gGZvPI8T2UGJG2FyCmKMYEGci8o6uvrZ20ALWXW6zJZi8sENh9I7uVUkAzM9vZ2+ntzJa8TExMIBoNF7dOebRrAsuyM5JVAp9PBYDDkTCUJtGkFoqTg2HgEZ7ZkW7fI4z3pi0NRFXTU2iBnkgKGg5NodlqQTEkwGrJGrWhSwO/2D+NT29rpFPH1k34IkoJL1tRRMvbse6PY1uGCVdPItX8oiE3uLHESJQWeUILmpwJA/0QMRj2H5qqsaazn3VGsa7LD7bTQ4/+f547jo1uaUW8zQUVa63rf04dx74fPoAY0fyyFZ3rH8Inz2iiJTwoSBFmB3ZR/4f7tO8PYuaEBRj0LVQXCCQHf+uNRfPujG+nvBCcF9HkjOKctNyOXkPxn3hvHxhY76u0megHxL785hH++eg3NeRVlBcOhBNqq05msU/NmH907AJOBw42bmzPyhbRuW9TEZel0OrS1tc36ZVrKrNeenh54PB60tLQgEolgaGioLCFYHCzbtXhychL9/f2QZRltbW0wm8201ru5uZnqqudKXkVRxNDQEERRLGol6mxyr0QisSzIq9frpVm682kbk2WZEjwSV1VRUUEns4UQKW3GqtvtphWwpYbW6T/1NSHRgaQYQlVVWgxhs9kWNfKS7NIWekFRyqzXZbIWlwlsPkxHYMkUoKOjY86yAe0iUco+7XzTAJZloShKQbl+HMdBbzDkTEhVVUUglkJ1pRGBeAo1lRWnT2NjPE5OxHDeahf9WTQpwmbSIxBP619Jy5WqAo/uPYlPX7A6Wzcb5aGoKurs2WxX+phkBaKs5JQp+GI8HBo5gJxpxVqlkQxIsoLxaBKNDnM23SDPschjZBgGvaMRrG2w5Tz+X7w+gJu3pnWkDIBDw2H8dv8wvn7dGfS+j4xFMBpK4nLN9n08JcGkY8FxLLxRHnUaw1tKkvGzV07iby/rpNPjo2NR/M+hUfzjlWvoY0gIEhgVMBl18MV4OC3GHCnB137/Lu7Y0QWHyQCWScspet4bo7W15LERWQgAyLKCVIrP2bor1CgwODgIURRRU1NzmrRmseHxeOhFGADs3r0br732Gn7wgx8U9X7zoExgS4Rjx44hmUyira2NTuOm1nrPh7wODg5ClmW0tLSUhCjlk3uRrWkiG1gqzSsZqjidzmmzdOcCEuMUjUZzJtCE5OUj6aTpi8jZlipjlRjHCkk9mKkYghQmzPe5JLGahe7SaiVcpch6XSZrcZnA5sN0BJZMLTs6Oua0YGq3ZxZrkZgPkskkPB5PTsRXIeYvlmXp1dyQPw53TVbbms6LzZKhpCBh/2AQ2zqzRQNEchCI8YinJDQ7LTQkv+fdUexYV59jfgLS2/Nj4STqHSZKvCaiPPac8OHaTc1pvaai4sUj4xBlhbZ0qaqKH73Uh50bGtBabQXDpCeo33vmKP7lmvU0EWA4lMB/vTmIf7pqHb2/494YTkzEcNUZjTQlYDScRGWFLmeqygsSTQ2YiPI5yQt93hiOeWO45sxGStB//ZYHvriA2y5eTfWrBz1hNFWZckisKCnQ69h0zNdQGFtWOekx9vYH8Mx74/j6hzfQ8x0ITCIhSFjfaM/7fnr8rSF88IxGmPQcVAChhIDvPnMU37wxO/GVZAVCiqfTD1EU0draOuvVOzEW8DwPu92O1atXl/w97fF4sGPHDuzbt6/UW1llAlsi8DyPVCqV836UJAlHjx5FQ0MDqqqqaPJLIWuxIAi0BnSptqi1MjKChZi/FoKJiQkEAoGiDVWmc/iTCmoyTFkOTV/kvQFgzsYxWZapnGKh7V/xeBwejwcWiwUtLS2zviZaCVd7e/usUoNiYInW4jKBzYfpCCz5kK1evXpO5JXoUua7PbMYmKp51el0czZ/GQyGnCtSQlDjvAiTQYcT41F0N9pPm5hGkwJ6h8M4v7OW/mz/YACbWqtPI8CyouK53jHsPCNNAFWkSeKTB4bx+Us66TT3zZMB6DgGm1uzsoVne8dw7qrqnIKCZ3vHcOmaOhrVBQA9745h5xkN9HeOj0dxaDiMj5zjpukDv37Lg0lBwl9fkG3peuz1AbTXWHOC/1/v92NbR7YjXVFVJAUZFqMOvCgjKciosmTP58kDIzDqOVy5vp5KTx585STWNtiwrSN73KHAJDWykSpbcowjoxEcHI7g4+e6qRnuxSNejIQSuGXbquzElxdhMnDgWBZHxtIpCYSYswwDRVHA83yO8L+QhpmpMTvd3d1Fz4eMRqPYsmULnn/+eWoaILc98cQT1FBQIpQJbIkwW6230+kseC0mhRyqqqKlpWVJ9ZUjIyMwGo1obm5GIpEoetZ3PpAt52LK2QjIBJqQWUmSqCmK1FQvZdOXVntbyAX8bMfK1/6Vr+J1KpLJJAYHBws285GJsaIoaGlpKbjmfiFYRmtxmcBOBxJurMXw8DBisRhWrVoFnU5XEHklIfBL2aet1RZN3Z6Zq/lLS2IH/XG01lgR50VYK/SQlHT7lba+9Y3+CVy8poEST6J/FSQZyZQMq0lPTUWPvzmIi7pdaMgkG3Asi9f6fLigM6v/EeV0CL9Rx0LImJbozyQFKSlthiKI8yIMOhYGTdJALCnCUqHLkQUIkgyDjgMvypBkJUdf+8bJAJqrTGiqyiYuPPDicWzvdOHM5rQGj2UYfP3J93DHji7YTXqwmbSGvf1+XNSVzVtVFBWKmtYC9/visJv0adNX5rjfevoIzl9djQs6s4UCu98cwoc3NVGZg6qqSEkyKvQ6BOIp+GKpHMPWL14fxESMx5d2dNFJ9UlfHDVWA6oyrWla8kousApt2xkfH0coFKJZr6WIuIlGo/jJT36So7Pq7e3FDTfcgGPHjhX9/qegTGBLhJlqvauqqmhb1mxrMc/zGBoaAgC43e4l64In5iBikNQSlGJmfU8FichbTCNxoVBVlZL2SCQCIL0b6HA4Fl1HWgi0w53F1t7mq3gl7V+VlZU5FygkOoxl2YJyb6dmvZYqDWAZrcVFfZNwd99990w/n/GHSw2yzU5AsvAmJycRDochiiI4jpuWyCqKgpGREcTjcdTW1i5L8gqkFw6y1eF0OmE0GiFJEiKRCEKhEO241+v14DgOsiynnxuGhcnAQc+x4DJ1szqWhSgrUFVg/1AA7morWpxWsAwDUUrLCCr0HE6Mx+CyVVBC9uKRcViMOmzNGLMYhgEvyNDrWDgtBug4NmfC+epxH1a5rFR2MBxMYPebAzhnVTWMeg6youKV4xPY0+/HJreTJiY8vm8IgqigJTPZVBQVj+w5hXWNNkpwJUXFq30+mkYAALWVRgiSQs8NAM5dVY1aW7py9eBwGFVmAy5fV48KHQuWYXD/M0ewuaUKHXVpg1wgnsK9f+jFFevr6XENHIPxaBJOi5GS6e2dLrir03m1fzg4ArOew/ZOF1gm3dz15d8cwtoGG6rMhnSDGsciEM9qYVOSjE3uKmxtr85k6Y7h3ZEwLux0wWTQQVbSuwvj4+NgWRbBYBDhcBgul6ug9AC/349AIACO49DV1VUyImA0GnH48GGcffbZ9LYHHngA559/Pi644IKSnIMGXy/1HRYZdy/1CUwHUg4z9bZQKIREIpGjPZyO4BHySjJFlyN5BdJSLZPJhKqqKtjtdnAcR4leMBgEz/Ppz7xevyCCFwwGMTExsaAIx4WAPIbJyUnwPE+32KPRKMLhMMLhMGRZBsdx4DiuqOdHqoOTyWRRtLccx8FkMsHhcMBut0On01EtdDAYRCKRoBFdw8PDUBSlIPkCId2iKMLhcKC1tbVkr+MyWouLug6v6AmsIAh0i5eYBGRZRjKZzHEg6vV6Oq0kbzptn/ZSVs/NRl5nwmzTAJJQQBbh8UgS9XYTxsIJNDjMdCuefKgEScE7A35s7ailz+t7I2G01VhRWaHPbfZSVfSNR2E2cGisslCT0v99sQ+3XrQaOo4BkJYuPPHOMHZtb6fTxwNDQfCigq2ra+gE9elDo1jTYEO7y0onnY/uOYUd6+rR4Mi6+b/59GHc+YGsYWoklMRPXu7Dv96Q1ozKiooTEzG8eMSLz13ckc2YjfOQJBX1juyWpNbU9tZAEJtbq2g7Vigh4oEX+/DVD62nMon+iTieOjSKO3Z0ZVMNxHTChdmQNmxVWQzQZS4WGIbB15/sxa7tbWhymMEy6Yn3I3sH8bmL2nMIv6wo0GdeJ1VVcerUKYRCIXquRqMRjY2Ns36xk2xYhmHQ2dk578zMheDBBx+EzWbD0NAQHA4Hbr311pKfA8oT2JJBG2mYNh+qdCpLJltkfbJYLHR9IpIWUkVKplpLVaE5G3mdDovd/EWaHwvNdy4GtMax6upquu2dT0da6Nb7fM+D7JAWuvu0WMinDQaA6upqVFdXz/j+KFXW62xYBmtxWUIwHQiBnc7hKssyjaoiOheTyYTKykrEYjEkk8mCstuKhcXMOJwpCqaqqgp6vR4joQSanRbImWkJl5nGsgywfzCIc1bVQJIVcCyD0KQAs1EHA8cikhRgMxly8l5D8RQ+eFYzJYF/2D+My9fX0ypZABgLJzPkM0t8Sa0sL8gwG7O/KykKBEnJ+XtZUaEiPTWeGjMFpBMA1jRkxeiBOI/XTwbxwTOzlbjPH/bi8FiEZszKiopnescQSYj4qy1uKgHwBBKorTTCZMy9f45lkBRk7D3px2WZmDCGAfb0+/HUoTHcd8OZALKGrUA8hbPbnNl8XVWlDWdPHRrF9k4XbBXp4gVeVHDXb9/F9z++id6noihIpVJQVZW6XMlUHZj5y4JEyxQ763WFoExgSwRCYLVrMZlYkc8qaU+KRCJUW1lZWQmj0Qifz0cd5cXUk86E+ZLXqZir3GsqQqEQxsfHaR36UpBXINvYN5NxjFygRCKR07bebTbbghMbtMbqQmqyiwWSvpBMJulaTL5fSWGClpxqz7vYWa8rAGUCOx0EQYCiKAWlDYiiSBdQcjVlNBpRU1Nz2huwFChWQPd004C6ujrU1mWTBMKTKdjNBpzyx9HuqqRxU2xmi1vHshjwx9DuqqQ1qr0jYUSSIi7sqqVEciSUQFOVGZKiQFFVGDILv6KmW6naXZUw6NL3KUoKfv3WEG7a2kbPdyLK41dvDuCOy7vTiQmKiiNjUbw1EMAnz19FSfNBTwiKqmKz20lTAo57o2ivsVKNraqqiCVFVJrS23dJQYbJwNGJ7tGxKGRFxfomO73tJy+dgMOix1+d46b3/6s3B7GuwY6z3A5621gkiQa7CSzLYDIlwaIhukdGIzgwHMFN57ppgsLzh8dxeDSaM60NJwSY9CyMel1ORJeWlPM8D0VRqFHAaDSitbWVXoxpdVoVFRWw2+2IxWKoqamh21vFznpdISgT2BJBURQ6TNCmp+Rbi7Xaymg0Snd6HA4HqqqqlkQ6sFjkdSrm2vxFLlgtFguam5uXZGIHZCfAc9HekliuSCRCvSkWi4XqSOfznBIDm3YCXGpoozXr6upQVVVFd3i1369WqxUGgwGyLEOn09GLsmJnva4AlAnsdEilUpAkaV592haLBTzPL3i7Zz4oFnmdiqnTAL1ej/b2dlgsFrw3HEJnnQ16jqWGpZQoQ8+xODIWxvqmKpp1Gk4I8EV5rGm00+eaIJIU0DcexabWbJ3q7/Z7UG0x4qLuWppe8O/PHsWnL1gFu9kA8pb7zz0n8Zntq+lU88REHO+NRHD95mYqN3j60CgUqLjmzKZsicHzx9BdZ8OVG7Ik7ds9R/Hhs5rQkXHxi5KCx98axM1bV9HfEWUF45EkmqvMlJQTnS3LMvhznw+tTjPc1Rb6s6/9/j1ctrYW2ztcWSPbWx5cd1YTTIYsWecFCWajHpGEAF88hY7aSkpMf/XGIMYjPP7+A9kp8HsjEXAsgw1NaUevoqgQhBQlA8Qo0NbWdtokQxAEuoAeOHAA9957L6xWK84991zs3LkT11xzzZKErS8zlAlsiSDLMgRBmJW8akFiiDiOQ0VFxWkEj2gRi41ikdepmE3uNTk5iZGREZjNZrS0tCwZeSUkeiHyBTJtj0ajObuB+aaV02E+JLoYIJNop9OJurq6nJ+RizGyFj/wwAN49dVX0dbWhvPPPx8333wzOjs7l+S8lxHKBDYf/H4/vVolxHMmSJIEj8cDnudpjl2+7Z58etnFRKnI61RopwEVFRVobmmBLrNYx5IirBU6DPjjaKoy51TJjoYSaKwyI84LMBv0VHs6HErgj4dGcOvFnXSb/OVjXjRVmdFZZ6MTzoOeEM5odlBtKcMwiPEiKiv0EGWFRmeRHNqUKNOJKvm5pCjQsSyCkwIsRi4nrur1fj82NNlzEgkefKUf153VBFdlOjlAlFXc9btD+LcbNqa39pm0YeuxvQP40hXddLufF2WEEwLq7en3EnkM5LiP7xvCllVOrKqxUoL7j48fwN9d3olmZ7p4QZQVHPSEsMntzI3Dyvz3tRM+RHkJV21ooPfBIL3oK4oCSZJoa1ZbAUUF4+Pj+NrXvoa3336bEoiuri48+eSTi77o79q1Cw8//PCiHrOIKBPYEkBVVfT19UGSpIK3jklElcFggNvthk6ny0vw8ullFxOlIq9TQbSVRO5FYDAY0NY2e7NesUD084s1AZ5uN5DUu1oslrxrFDmPpZZRkOKBQox0qqriqaeewo9//GOMjIzQ2++66y7ccssti35uK2gtLhPYfPD5fDRyBZh5a6YQ0jidXpYcczEWlaUir1ORSqXg9XrR3t4Og8GAN/p9aKoyo6nKTKtkA/EUqq1GeCNJ1NlNtBkqyov446ER3KSZbHojSdjNhryxWZGEgBgvppu1MhPatweDqLcZ0ey00AipX781hOs3N9OkgcmUhMf3DeKvL8iancIJAXv7/fjA+ga6LR/jRcSSIhqrsrEqRNfLsSw1rmnxm7c9uPHsFiiZCXPfRAwPvXoS3/rIWTSz9ZQ/jj8dmcBnLmyn98WLMqACFQYO3iiPamuuYesbT/Xik+e1oiVTAsGLMn7wfB/++eq19L5lRUUiJaHSlH2PplIpKoMZHBxEKpVCa2vrrBdl2qxXWZYxMDCAnp4ecByHH//4x4v6Rbh792589atfXYo4rPmiTGBLhMOHD1NZi1Z7b7VaT/vSJ+RkJtI4nV7WbrcvWqf9UpFXLVRVRSAQgM/no7eVejeQgEzETSYT3G73ol8wkGERMVcrikLrXe12OyoqKsAwTNHPo1BMTk5iaGio4Ik4yXoluxGvv/46XnnlFdx888346Ec/uqjntsLW4jKBnQ6pVAqBQADBYDAnE1Z75T40NISRkRFUV1cX3KdN9LLRaJQel1w1zlcvu1zIqzZPj5yH0VgBjmhjEwIcZgPGIwlYjfrMNjkDjmXwfO8YdqxvOO2Yoqzg4FAQm9uqM+H7Kt4bCWPPCT9uuzS9haKoKh7bewoNNhOu2JDNnH3gxWP40MZmtFZbqNnr+88dxd9emtWOTsR4PLrnFP73VeuoVndvvx8vHvXiKx9Kt16lG7+8OO6N4rZLOqmc4b2RMIw6Ft31NvplEEkIsFTooGNZOg0m8MV47O0P4Nqzmqjh7E9HvXjhsBf3ZdqxFEVFvy+OiSiP8ztq6ARXVhQwYMCyDF4+NoFz2pywGNOxWJKi4H/tPoBv3XhmTgYu0bxqyWhzc3NB6QGlynqNRqPYvXs3vvOd76yURRMoE9iSQVVVxGIx+P1+RCIRGqmlJShGoxF79uyhRLSlpWVW0qjVyxLSQwxRNptt3tpCQqJNJlNB51EsxONxDA8Pw2g0oqWlhU4r52P+WghIS1ShwfwLxXT1rmazGZFIhOr+l+p1IWRUr9cXdB6lzHpdgWtxmcDOBnJ1FwgEEAqF6FbqoUOH8N3vfhculwuPP/74nLuHVVXNmQbMVy+7nMirx+OheXqEJDEMA6PRCJZl8Xq/D7YKPdY02ql+9ZAnhDNbqiBIaY0sdRZLMn75+gB2bV9N9bLDoQTeHQ7hmo3N9H7fGw6j2WnOad/SmpiIrjaRkmA26iArCtiMji6SFGA3GSjhFSQFvhiPpiozPVafN4aEIGFjSxW97b/3DcFk4HDNxia6ff+9Z49hVY0F157VRAnu0++O4tw2J+o0soHwZApVlnRHdmhSQJUlSwzfGQyibyKOj21x04ayJw+O4Nh4DHdeuYbGiXmjKZgNLGwmAwRJoUY2LQRBgCRJOa7VQlMxSMg5x3Ho7u4uamPR7t27sW3bNuzYsWOlLJpAmcAuCRRFQTAYRDAYRCwWo7f99Kc/xSuvvIJdu3bh9ttvn/MQgJCeSCSCeDwOYH562eVCXmcijXM1fy0EJMKMpECUQnesBdn5DIfDdIpvMBhoYUKp3fuEjAJAW1vbrPdPYjBTqVRJ6rpX4FpcJrBzAYkfevrpp3HfffdBkiTs2rULV199dc40YD7HnY9eVkteC50AFwOFdFnrdDpwOh2NvBoMxNFabcVkSsxsnesoQXv4z/342HltsBiz0VYkNkuUZZppCqQntKf8cax2VdKp6ngkid7RCC7prgXHsjRpQJBkbFmVrWt9+dgE1jfaKMGUFAVv9AdyqmElRcFYOJnOWiVtWqoKJSOHGA0nYavQwVqhp2T2l28M4IwmB85odlBy/E+/Poi/2tKCs1oc4FgWkqzgt/tHcMPm5mxLl6oiKUiwGPWIJkWEkyLczqx84bG9A4inJNx2SQcAZAobfBAkhZrOFFWFqDG9EKNAoW7bUma99vb20vfKClo0gTKBXXIIggCv14u7774br776KpxOJ772ta+htbWVkrH57mbNRy+7XMgribvTaoCnQzGbv0h5RKGtUsUCIY2KosDpdGJycjInlkuba15MEDIqimJBZRrarNdS1HWv0LW4TGDnilgshm3btkEQBNx5553Yvn071bUC6fghh8Mx7w9FoXrZ5UheZ+uyZhgGBqMRHMvi8EgYoqzgjOYqMJnCgz8eGsGla+tRoc9dOBVVxbGxKNY2Zo8dSQj42Z9P4B+uXEfjpfYPBbG334+/u7ybGpieeNsDg57FhzZm0wd++GIfNrVW4fzVNZRg3v37d/GJra3ocFVCRZoc3vXbQ/jatRtQoefAZMxZ//fFPtx93Rn0PMIJAfuHQjl1saKsQJZVVBg4BOIp2M36HD3rw6+exKVr6tBabYaiAhzL4P/7r3fw5avWwWVLt3KlRBl7TvhxUXf2uOnnmxQ2hCDICs5dVU2JM5CdvAJZ12+hbttSZ73u3r0bH/vYx+DxeFbSogmUCeyywGOPPYZ7770XjY2N+Ld/+zdUVFTQ9z7DMFRSMJ2hZzYUqpddTuTV4/HMa+I51fw1m954JpCkE9J8tlTlEZIkYXBwEJIk5ZBGkrSijb0spqlP+x3pdrtn/a7W7poZDAasWbOm6BcAK3QtLhPYuUJVVXz729/GWWedhSuvvBJAequE6GWJ85NhmJwPxXwW0Kn5smRRsVqtCAQCEEVxyckraRybS5MJx3HgdHqaG/v2QABnuZ1gkE4MMGQIrKwoeOa9MbS7rOisS18dciyDn71yAru2r6bELs6L8IQSWNtgpyQxlBBg4FhYjDoqI4gmRViMupx4q3yZqYeGw9jQaKcTVwD4w4ERXH1mI8hNY2Eej+w9hX++eh2VOLw9EMTjbw3hO3+VLg9QVBUD/kkcGYtg54ZGer4pUQbLAnqOw3AogcZMBiy5/2/98Qg+fUEbXNb0ecV4Efc9dQTf/uhGej6ipGAiI3cg0JJXYliwWCxoaWmZ9f2XSqWoUaCxsRENDafrkRcTPT092LlzJwCstEUTKBPYZYFjx47hl7/8JW677TY0NDRAVVVEIhEEAgFEIhGaA6vT6XIMPXPFTHpZjuMwMTGx5OSVbNdzHLegiedCm79EUcTg4CCtRC2mvnYmkIIAnuen/Y4kMj5CZiVJohc+xJOy0C17VVUxOjqKaDQ664CHgOyacRyHNWvWFD3rdQWvxWUCu5hQVRXxeBx+vx/hcDjHcECu3Ofj/iSLCvmgke1hq9WK6urqkjpKtedEjEHzreEzGo3gOA7+GI/xSBJrG+10y/+Jt4fQVW/DxpYqOmE8PBrBukb7aTW1iqpSSQKZRKZEGS8e9eLKDQ30Nn+MxwtHvPjIOW4qVxgKJtDnjeHSNXU5t3EMg2bN9n0gnoKtQg+9Rm9K4rl4UcZQYBJd9VnpxCFPGO+OhHHz1rZ0IgED/G7/CA56Qrj72g00K/bYeBRRXsK5q7INW4KkQMemDVtvDQSxscVBM3VToox//s0hfPOGM1Fh4LKPN5M2AGS/zOZiFCDbWzU1NXC73UV9P3k8Hng8Hmzbto3+ewUtmkCZwC57yLKMYDCIQCBAtZ5AeoeMRHLNh+QRvWw4HKbHZRgGNTU1cDgcJdd5Allj0ELJ61RMJ22bTi6nnXi63e6iaudnO2+y/T6dpC3f3+QrESDvlfkmVHi9XgSDQbhcLtTU1Mz6+yRei2EYdHV1Fd3TssLX4jKBLRYURUEoFEIgEKCGAyAtIicLwFy3VsgCIQgCzGYzkslkSfJlp0LbIV1XV7egrWaWZaHT6aHLRFy9cHgM6xrtaHCYc7bGgXSzlqoocNlMNDbrl6+fwvbOWrirzWDAgJdkfO+Zo7jrQxsytbYMBvxxPHlgBH//gTXU/f/S0Qmc9Mfx2QtXU7nBf+45BUUFPrO9nT7O/3jhOBrsFbjhbDed3D62dwCbW6uwvslOiehbAwFsaavO0cmSAoKkKCOWFFFry15JP3d4DHFeThcrZD4nj+4ZQHBSoA1biqLipD8Om0mH2srsl4G2XQtIE1Ay+ddu3xViFCDxWjzPw2azoaOjo+gXQz09PfB4PPTf4XAYP/vZz3DnnXdi/fr1dDFdxigT2BUEnufpDpm2d95iscDhcMwr/YXIBvR6PViWzWmIKma+7FQQrWmxt+tnM38feiXKAAAgAElEQVSxLEu/m9xuN8xm8yxHLA60E8/5VrlPV9lLyCyJ5ZoNpDBhpspcLYj5TlVVtLe3l6SGfoWvxWUCWwoIgkDds0RADqS1rUQvO9uULJ/mtRT5slOhXSAWSl610Ov14DgdFFWFJziJtpr09o2iqjg0FMLx8Sg+em4rgPTE4+ev9uPStfVorbZQovvn4xO4oNOVQ3qJ+YtoXSVZQVKUUVmhp9Fa4YQAHcvkGLEOj0bQYDehymKgt/3mbQ/OaXOm7zMjQfjq79/FDZuacGZzFcAALMPgn399AHfs6EK93UT1rD3vjeFDG5uyBFdRkRRlWIw6hBICRElBra2CktOfv3YSFXodPn6um/7+f705BD3H4MazW8BmigwkUaSyAZLXKkkS2traZt2+007RTSYTuru7l2QLtLe3FzfccMNKueoHygR2RYIQE7JDRnYsiLbV4XAUtJuVT/NainzZqSCyn1JrTfOZv1iWpVXThUw8iwFtNWuhE8/ZkC+hggyhbDbbtGvsXAsTyBR9qeu6V9haXCawpUYikYDf70coFCrYcFCIYWs6vex8RPjTQVVVjI2NIRKJLNoCMRUGgwEsl94af3zfIIwci+s2t2T1qZ4Q1jXaodO0aBEE4ynoOAY2U3ohlxUV+04FcEazAxZjemtPlBT8z6FhXL/ZTf8uIUjoeXcMH97cTMnvZErC0bEIzsq0XgHpaK+kIMNhzv2iIAT3ncEg1jfZc9q87u85ik+f34aaSiNUpPWsd/32EL73sc3Qc+k4rzgv4uVjE9h5RlYnqygqZFWBnuNwbDyKCj2XQ9aB9HuJ3I9W81XoBIRkver1eqxZs2ZJzBY9PT14+umn8cwzz+Czn/0srr76aqxfv77k5zFHlAnsCoeiKAiHwwgEAohGo/T2mbbIgdkNWzPpZRczc1UQBAwODkJV1SXVmvI8j+Hh4Rzvx2J/7xQKbTVrbW3tot+3LMtUYkAGRvkkKcQMS4osZpvEa7NeXS4X3G73jL9fLKzAtbhMYJcKqqoiGo0iEAggHA5TwwEJ6HY4HKioqIDf78f4+Dh0Ol1Bhq18IvyFanDJcYkzsqamBi6Xa16PuxAwDAO93gCdjsupfwWApCDh6HgUZzZX0TisV455MRpO4qatbbSe9XvPHsW21S5sXV1Nnf5ff/JdfO7i1aizpdu/UqKC+3uO4K5MYQEA9E/E8OjeAdx7/ZlQFBUqgDdO+vH7/cP41kfT5ixZUfHOYBB7+wP428s6qXZ2JJxESpTR7sou3IIkg2MZcCyLPm8Mq2utORPi7z57FF+8tBOGjKFtJJTAN58+gh/dcg79ncmUhOPjMWx0O+jfer1eDA8P060tnudnjDGbilJmvb4PUSaw7yOIokj1slN3yLSRUidOnIAoirRBabbdinzb7hUVFXSHbL56WS15dbvdRTf5TAftDk59fT0qKirmbf5aKEKhEMbHxwuqZl0MiKJIyaw2gsxsNiMQCFD/wWyvcamzXt9nKBPY5QBZlqlelmxTAEB/fz/+9V//FatXr8ZPfvKTeZUlzCdfNt9xyNZMdXU1XC5XST5kHMdBp9eDY1m8cHgce/t9uOtDZ1DN6AtHxlGh43BhVy2dRL5yfAJntzpzMmRP+eJY5bLSx8IwDHwxHq7KCioFALJyA/I70aSIkXA63YDg3eEwvFEeO9bV07/9xesDmBRkfO6i1QDS09NfvDGAUEKgrV+kucuk53LMXklBglHP5ZjUCPzxFH7wfB/u+fAGKoEA0l9ggiCc9uXIcRycTidsNtuMr20ps17fp3i/fcOU1+IM8iXKAMAjjzyCnp4e/MM//ANuuummOUtt5psvOxXEcCnLckF5osXCTFKyuZq/FgqynhWauLLYmBpBBqTJbFVV1Yz66lJnvb4PUSawyw2kwva1117Dvffei1QqhU996lO48cYbF2QOmK9eVlVVTExMIBgMFm1rZjbo9XqwHAeGYcAyDLyRJIx6Dg6zAYIkw6DLnjsvyhgJJdBWY6WTUU8wgeFQAue1Z0sM3h4IoELPYUNzeqqZLgWYwMZmB5zW9AIrKQoODoWwqdWpqXRVEYinUFOZzmslSQRET3vKH4eeY9GsafP6+Wun4Ko04oNnNtJJ+3efPQZXpRGf3NpKkxfeGQyhxWlCvT07DZ3atCVJUo4RhUxSzWYzddIC6deWbG1ppwDarNe2tjZUV1cX4RV736NMYN/n0FbYfu9738Nzzz0Hu92Ob3zjG+jo6KCRXMXMl50KLXldapc/2Y2bTUpW7OYvYnwqdLu+WCD+A0EQYLPZkEgkcmK5psoDlyLr9X2IMoFdjgiFQrjsssuQSCSwa9cu7Ny58zTDwULMAYXqZVVVhc/nQyAQKNhJWSykUikYjUaYzWaEJlP4zdse/M2F6e0WVVVx0hfH7jcH8S/XbICqqlBU4KmDIxiLJPH5SzqpdvRHf+pDa7UFV5/ZSCeo9z3Vi8vW1GGrptjgf//3AXx6WxvWZKavkqLgy78+iC9/cB2cFiNYBogmRfyf54/jq9dmJQhJQcL+oRAly0Ca9EqyAqOew1gkCbNBB7spaxj70Usn0FlrxY519ZTgfuuPR7G2oRLXbGyiiQSSlDVsAdnIlcrKSjQ1NYFhGLq1FYlEcpzRhw8fhtPphN1uh6IoJcl6fR+jTGD/QvCLX/wC99xzD5xOJ+69994cZ/hCEmWAuelll0tEFQBMTEwgEAjMeTdusZu/iPFJp9Ohra1tyTJ4tZPUlpYWWK1W+toSmYGiKOC4tDfi6NGj6O7uhqqqJct6fZ+iTGCXI5LJJG6//XZccskluOWWW2iFLTEcaAO6F2IOmE4vSxblWCyGQCAAh8OB+vr6JSWvxG27evVqVFSYaOzWYGASr/X58ImtbfT3T/nj0LEMWpxZ05M3ysNh0sOo5yhxDU0KsJv1YJlsscFwKIFGh4lqaRmGwdsDQWx0O3IMY7vfHMK1m5pgyhwvlBDw3WeO4hvXn0nf2Cd9cXz/uWP44SezelZ/PIVX+3z4UIaYAuk8WVlRYdRzGPBPwmkxwGbS58RlTZ28kskDMZPkmzyQSc+pU6dw2223AQBcLhcuv/xyfPKTn0RnZ+eivD5/gSgT2L8QvPzyy3jooYfwla98BR0dHRAEgUoMCAkDskRsvhW2M+llzWYzRkdHlzyiCsgapRb6nbDQ5q+5xgUWC1pj83R56CQfPhqN4qGHHsITTzwBhmGwfv163Hjjjbj22muLnvf6PkWZwK40iKJI9bJTK2wXYg7QfsiIbglIb/c0NTUtmctVa1hoa2ujkw6tPnYwEEeL00K3+SVZwZHRCNY3Oai+NcaLePrQKD56jpu2Xg2Hknjh8Bhu2daennJmJrlHx6K4KpMIoKgqPIEEZEVBe21WKxqaTMFi4GDQZ59rkoogKyoODYexyZ2d1gyHEvjN28P40o6ubHPXYBD/79WT+MktW+jvjUd47B8K4gPrG7LVtJqcV2DukwdJkvDII4/g2WefpfEoHMfhySefREdHx9xflDx48MEHYbPZEI1G4fF4cOeddy5ZnE4JUCawZcwrUaYQ5JtUAkB1dTVqamqWbJuc5JouplFqPqbj5dL2BQA+nw9+v79gY/Pw8DB++MMfYu/evQiHwwCAc889F48++uiinE95HV7Eg5cJbHExXUA3uZKdb4Ut+VByHEelC8XOl82HQgwLHMdBrzeAZRn4Yzy+8T+9uO+GjTS5YDiUwKN7TmWKDdIlBm+dCuCdoRBuu6STtnr9bv8I4ryIW7atotPYH73UB6OOw2e2t1NC//3njsFhNuCvL1hFJ7e/fGMQDfYKXLqmjhYsHBgKoauuEpWm7GQgkZJgNuogKQqGg0m01WRNecfGo3jxiBdfuLSTniebkQRoySuJXAFQ0ORhatar1WpFT08PBgYG8OUvf3lRDFz3338/Pv/5z9OF8o477gAA/OAHP1jwsZcpygS2DAqSKOP3+xe1wlarqyRrcSnyZfMhHA5jbGys4FzT+aAQ8xdx7QuCgNbW1iWVUpDkA7vdjoaGhoLqugcGBqAoCurr6zE0NIRnnnkGXV1d+MQnPrHg8ymvw4t88DKBLQ3I9DQQCCAUCtEK2/nEmJCrbKKrlCSp6Pmy+TBXzZdOpwOn04HTTCeIq59sxcuKinhKgt2khygr0HPpaakoK6jQcxBlGXqOw2RKAsswMBmycoP+iRiqrUY4zAZ6vD8cGMH6JjvaXVb6e1/7/bu4uMuFS9fUQUWahP7LEwexc0MDLuhwURPZM++N4cIuFyor0gRUVdPnRv5NngPthQlZvEVRLNiBXIqs1xtuuAFXXXUVbr31VgDpKcDu3bvx/PPPL/p9LROUCWwZeTFdhe1cjUuyLMPj8SCZTKK5uRlWq7Uk+bL5oHX5Nzc3l2QCPJ35S5ZlSJJEtaZLhVgshuHh4YKTD7R13cXKei2vw4t88DKBLT1mCuieLT6LXFHmu8qeTS87X0duPsiyjKGhIaRSqYKyb7UwGAxUQvHInpO4blMz7JliA0lW8OSBYezc0AhzptggIUj4+Wsn8cVLu+gxIgkBv3pjCJ+/JJvJF02KeK1vAh/YkC0b4EUZwXgKjVVZTZqiqoCqgmVZHBmNoK3GCpMhW2zw788dw+Vr63BGs4OS3tt/8RY+vW0VNrc6aXKCJEk5k1dVVTE0NIREIgG3213Qc0IuRoqd9erxeOiXM1C+8l+BKK/FRQBJlAkGg9RQCcwen6UoCjwez7S5zsXKl82HeDxOtfZL5fInQxS/30+HMwsxfy0UyWQSg4ODMBqNaG1tnfU5Id9nPM8XNeu1vA4v8sHLBHZpIYoiXUDzBXRr5QBki6iQq+x8etmFOnIJSKMUmTzMZ4ubYRjodDrodDq6UPzTf+/H5y7qQHutFYqqgmNZfPPpXnzxks6MYQqI8RJ+9FIf/vdVa+m78+hYFI+/NYS7rzuDalf39vvxxNsefO/jmwEgE8sVxBNve3DfRzZSA9hoOIlj4zFc3F2bzXCV0pIMg47DcCiBensFdCw7o2FLm7nY2NgIuz2bFTsdlirr1ePxYNeuXXj44YfR0tJSkvtcApQJbBkFg2yNkx2ymRJlZiOvU7FY+bL5QCL3DAYDWltbl9TlT4xS1dXVYBhm3uavhYKYx1iWRVtb26wXC0uV9Vpehxfh4GUCu3yQTCap4UBb+2exWPCf//mfeOutt/Dtb38bHR0dc/qAzTdfNh+0H/bF6NRmGCZdTcuySIoyjDoOHJuubrVWpGUELANwLEslBSlRhkHHpiteUxKsRh3Nek1JMnzRFJqd2YzXQf8kPKEEtne66G1PHRrFSDiJz120mpYu/OZtD944GcD9Hz2LyhmOjkVxeCyCGze3ULOZIAg5UVlANram0Prepcp63b17N3p6enDrrbdi27ZtJbnPJUKZwJYxL0yXKKPX65FIJHDvvffirLPOwu23317QhaoW882XzYdkMomhoSHodLqCGqWKCbL+aY1SxWqcnAmSJFENayHmsaXKei2vw4t08DKBXX7QBnSHw2E8/PDDePbZZ1FVVYWHHnoIdXV1895qLjRfdrrzGhkZQSwWmzaOZL5IV9Pq6SL8nZ4jaHdZcN2mFqpJfeIdD2qtRmzvqqW37d43iHNanbQ5S5IV/PHdUexYVw+TIX0sSVHQPxFDR62NTllVVUVKklGh1yHGi4jzUk7D14tHvIjyIj68qRlKxrBFMly1sgEgKwMoNLaGRI7JsrxkWa933HEHHA4H7rnnnpLfd4lQJrBlLBjaRBmPx4N77rkHPp8P1113He64444FJcosRC9L1hCWZdHa2rqkAfukqGWmHPJSNH8pikIrXwuNMiORY0uV9Vpehxd48JVOYB988EE6fvd4PFQc/X7Bf/zHf+CBBx6A3W7HXXfdhaamJgBpHSlZ7OazeM1VL6vdIppaS7iYmEpkAeBHf+rDhV0ubGjKalJ/8PxxXLmhHt31NlpscNdvD+ET57VibYMNigpwLIMvProPX7qiGx21lWAYIJGS8ZXfHsJdH1oPp8UAhmGQFCQ88Y4HN53XltPmFZoUUFOZXVjzkVciAyjU+UsmBKIoorq6Gq2trSVxKUej0Zxp+Z49e7Br1y488cQTWL9+fdHvfwlQJrAlxvt5LU4kErjuuuswNDSE888/H1/84hfBZZoHtXKA+XyW56qXnS62cClAZG3aopbZIMsy4vH4ojZ/aZNcCpW1kZIZhmHQ1dVVEsNZeR1eXKzoUt/7778fNpsNO3fuxM6dO9HS0oL7779/qU9rUXHw4EE4nU48+uij2LlzJxobG1FRUQFBEODz+XDixAkMDg4iHA5T8XwhYBgGJpMJ9fX16OzsRHNzM8xmM8LhMAYGBnDy5En4fD4IggBVVeH1ehGJRFBTU1M08gqkFyJBEMDzPOTM47npvFZUmdOLNNnG/8KlHWh1pq+wyVT169dtwJr69JeIL5bWmv3wli3ork9PXl855oNex+D7N22m5PVbfzyCeErCTee1QVUBX4zHZx5+A5GkCKc1+8UgCMJp5DWRSGB0dBQmk6mgxVtRFAwPD0MURVRWVpaMvPb29mLLli3weDz0NrL1GYlEin7/Zbz/8X5fi/1+P0ZGRnDllVfipz/9KdauXQun05mWMcXjGBkZQV9fH8bGxpBIJDDLYCgHLMvCbrfD7Xajs7MTtbW1dM3t6+uDx+OhTVGiKFLpkdvtXlLyGovFqCdjLpmzHMflPF4ytZ2YmMj5PiM65NlAZADxeBx1dXUFkdfJyUmMjY0BSEcdloK8ltfhxceKnsBu2bIFL7zwAr2iiUajuPzyy7Fv374lPrPFgyimq0mnSga0hgOix2RZlsoB5hvQnU8vq9PpIEnSkrR9sSybjt/KTDv+/bmjGAvz+LePbKQ61ZePT+CdgSD+/gNrqLTgncEghgKT+PDmFlp2MBRIgGOBFmc2HSAhSDByLDiOxVg4iQbH6dKMfJNXsoXHcVxB+jOt/MJkMqGrq6tkmrVoNIrvfOc7OdtU5fiWFYfyWrzEiMfjp62riqJQiUEsFqO3L4ZhNp9eFgAlr0vZ9kU0/EajEW63e1HMY/Nt/iIygOrqatTW1hZ0P0Qn29TUhPr6+gWfeyEor8NFOPhKJbAejwc7duygrUUE3d3d7+dx/GlQVRWRSASBQGBRA7qB9NRxfHycbvOU2k2qBcuyEEUJlZVWyIoCjmXR894YzAYOF3XVUu3qo3tPocZqxFVnNFK5wX3/04sGhwm7trcDSD9n333mKEx6HW67tIOS3heOeMGxDC5fWwcgHbc14fVCUZScKBitUaDQLbxSZL3OhN7eXuzZs4f+u9wAs+JQXouXOQRBQDAYzJso43A45h0nRRJlRkdH6S5bqfJl80HbMlgM89hczF9EwlBo81gpsl5nQnkdXlwsnW1xgfB4PNO+6H9J43iGYeBwOOBwOE4L6CaLKdFTVVZWzklfNDk5icnJSVRWVsLpdCIajdJYrmLly+YD0TdFIhE0Nzejpibtcr2w04UDnhAUVaXa1U+c14ZJIZsQIMkK/uWa9VAy8VrHx2NwVRpx5861lPQ+tncAXfWV+MD6eppIAADRSAQTExMQRZFm79psNvj9fkiShNbW1oKIaDAYRCgUAsdx6OzsXJJtv/Xr1//FEIkySovyWpyGwWBAfX096uvrkUgkcuIRk8kkXUMcDsecdshUVUUgEKATQ+3QIhAIFC1fNh8EQYDH4wHLsnC73UW5PyJvM5lMqKuro+avSCSCcDhMzV96vR7j4+Mwm80FkVethMtuty9JdFV5HV5crFgCS8jUVJCO4b9EcBwHl8sFl8uVU2HL8zx4nofX64XFYoHD4YDVap0xiisSidAFl+g7zWYz6urqqAA/HA4jFApRQ5nNZisKOfP5fFR/W1lZiVSKx//f3v3HtH3eeQB/++sfGIO/toMx2ME4/Ejchea6dU1VMe3upuZ6pNWkNO2FKF26VOqWqZqi6pTdNnWiv9S0Gt2mTdp1Ez1VyaYQV1XUP6rhTVlPUa9ee9luWlNnQUmbgCHBgGPztfFPbN8fyfPEJhhs8A+wPy9pmoDK/rJVHz88z+d5f2QyGeqUSvR2GSGTyeD89DpOuK7g5Ld7IaqVSKXTmPCH8Z//fQmvPnYP7501auvw/t+neGsBAOzb2Y65yM1MV7YQjsViUKlU6OrqQiQSyVq8AzdzHNPpdFY27FIkSYLX64VMJkNXV1dFxyoSUgpUi++k0Wig0WjQ1tYGSZLg8/kQCAQQDAazNgD0ev2yJ2Rs0RWJRLJiC3U6XdYERq/XC6/Xm3VCVuws04WFBXg8Hn7yVI7kA3bq19jYmHX5y+fz8Z83NDRgYWFh2edhLVzRaBQajQadnZ1lPUEkpbFhF7C5/uJffMuvVqnVamzevBkWiwXz8/M8kovtqi43wjYYDOLatWvQaDR3XE6SyWTQarXQarVIJpP8w2tmZgYzMzOrzpfNhe0yGAyGrHxVdtmL9cg+1NOKvrtvxlFN+MP4n0sz2H+/DcceuwcAcO6KD+//3YvvP7wde+61Qgbg7KgXp/7XgzcO3odm7e0PkVgsxi8QsIV7fX09/33VajXC4TDGxsaWjYJhl7wAwGazlW1QASHlRLU4N5lMxo/6k8kk75cNhULw+/3w+/2oq6vjNSRzEcaGo8zPz8NsNt/xv6VCoUBTUxOampoQjUZ5/2goFFpTvuxS2NjcRCKB9vb2srctALcvf2k0Gly5coVHcbHPnuUmf3m9XoRCIahUqoJz1Mn6tWEXsMsFSBcaLl3NMv+CXTzCNhAIZB3J6HQ6nDlzBpFIBF/+8pdXnPYll8thMBhgMBgQj8d5AZ2amrpjN2A1BTQQCGB6ehparTZnvmAqleJ5tiqVCnK5HBZ9PUzaOj7cAAB2djTh7s03/71IpdOYDcbwT/YW/OO27Kb/zMVrptnZWUiSxIO6F+8G+Hw+HgUTDAbR3NyMyclJpNNpWCyWsg0qIKTcqBbnRy6Xw2g0wmg03jHCli3CWCRXKpXC8PAw7r33Xmzbtm3FzG21Wg21Wo3m5uasfNm5ubk1562yXeBoNAqr1VrRy2Ns5Gs6nYbNZoNarc66/HX9+nV+cshOAwVB4C1c3d3dFc3MJcW1YS9xAbVx87VUEokE75dlFw5GRkbwm9/8Bt3d3fjtb3+7qmJXaL5sLsFgEBMTE3mNzc3ERtTK5XIIgoBkKo1D//UR/v1f7fiHNgPkggyJZArDH49h9w4z33lNp9OIxWJLRpGxiwI6nQ5ms/mO588cFfmXv/wFr7zyCkRRxAMPPIBHHnkEDz30EP3FX37Vdj5ItbgKsYEGbAJjMplEJBLBsWPH8Nlnn+HIkSM4dOhQWfJlcz0fS0/Jd0R2qbAR5mwh3dDQkPXzdDqd1e71s5/9DB999BE6Ozvx1a9+FU888QRsNluFnr5mlbQOy1944YXlfr7sDyttbm4OMzMzuPvuuwEA7777Ltra2vCVr3ylqO/jcrkwODiI3bt3F/V1K0kul6OxsRHNzc0wGAxwOp341a9+BZVKhe985ztQKpWIRqN8ZzPfAsoGETQ2NmLTpk1Qq9VIpVJ8x5flGbJF5lLC4TAmJiagVqthtVoLbkVIpVK3R73KZHj8vnaYdfUQZDK87vw72gwa/PNdLahXyvnvFY/Hl1y8sozHhoaGnFmvgiCgvr6e70ZfvHgR4+PjuHz5MpxOJ86ePYtHH320aHPKh4aGcPnyZbz99tv485//XPR/36vEi5V+gCJ7odIPsByqxavD6qter+enTM8//zwuXrwIu92Oxx9/HKFQCMlkEgqFoqBLUzKZjC9Y9Xo9FAoF31xgdyNYvc41PWtqagqSJKGlpQUGg6GYv3pBMtspLBbLku1Yiz97UqkURkdH4fF48Le//Q0nT55EY2MjvvjFLxbtuagWr6ikdXhD78ACt6e/zM3NYXx8HN/73veK9tos8uL8+fMIBoN46623ivba60k0GsV9990HAHj99ddht9uzBiMUI3Egs1+W5ctqNBqIopjVL1uKiBalUsmPjS5en0Orrh76W4MRUqk04vGld14jkQjGxsbyzjrM3K2Ix+O4dOkSfve73yEej2N4eLgofWODg4M4fPgw3+kaGBhAIBDAL37xizW/dpWhHdgyo1q8didPnsSLL76I7du34yc/+QlisRivl0DhO6hLyeyXZfmyWq0Wer0+6z7EzMwMZmdn885XLSWv14sbN27AZDLl1Y7Fsl5ZrrnL5cL777+Pffv24fDhw0V5JqrFeaEc2EpzOBxwOp1VWzQB4MSJE9i2bRseeOABAKUN6M7sl2X9q42NjWhoaMDMzAxkMllJbrnK5XLI5QooFLfyXJMpJOKxJafmxONxXL16FYIgYMuWLXl9WLAiW8qs14GBgawgbLfbjb179+LcuXM1f2FmEVrAVqFqr8XXrl3D6dOnceDAAT7xMDNRJh6P83+W3TFY7Qhb1r7A2r3YpShWR3w+X862qXLy+XyYnp6GwWDIeRciU2bWq9FoLFnbANXivFAOLCm9J598MutrQRD4Ddd4PJ4VyTU7O4vZ2Vl+61MUxYJ6PFUqFYxGI789ywooWyiLooiFhQUoFIqiFs5kMnnrP3LI5XIkEoklF68sLgYArFZrXotXlrkrCAK6u7tLsnj1eDxwOBzo7+/nWYKsUM7NzVHRJGSDs1gs+O53v5v1vcxEmVAoxCO5QqEQTxzIlSizHBZB1dDQgNbWVt4vyyKqBEFAXV0db1+oBEmSVrzImykz61UUxZINKqBavD7QApasSKVSwWw2w2w2Z104CIfDCIfDmJqa4pEthQR0s8BqlUqFcDiMZDIJjUaDYDAISZJ4vuzieJm1YgvZpWQWwHzjYoLBYFbWa6lu6Vqt1jsmG7lcLoiiWJFQbkJI+WRGGC6XKMNqZiF/RBGKwz8AAA5NSURBVAuCAJ1OB7lcjvn5eSiVSgiCgOnpaUxPT5c0XzYXFkNYX1+f16CCzKzX+vp6dHV1lWznmGrx+kALWFIQjUaD9vZ2WK1WSJKE2dlZvoMqSVLBI2zZgjEWi8FqtfLAatZikE/GX7GwAshCw/NZiIbDYUxOTgK4mfVa6r+8F09xGRoawssvv1zS9ySErC+CIGDTpk3YtGkTEolE1tQvdkJWaCZ3JBLBxMQE6urqYLPZIJfLs07I1rLbW6hYLAaPxwOlUgmr1ZrXorncWa9UiyuPFrBkVRYHdC81wpblorKxf4uxm6XhcBgWiwWNjY0AcufLZmb8rSVfdinpdJoXwJaWlrwWorFYDBMTE0in0zCbzWXPeh0YGMDRo0fR19dX1vclhKwfSqWSj7BlC1i/389H2OaTyc0WjAqFIuvCKsuXNZlMvF928UjX1ebL5pJIJDA+Pg6ZTJZ3Co3P58vKei33uG6qxZVBC1iyZpkjbBcHdLMjKBbQrdVqIQgCj2gJBoMwmUw58wWX65ctRjoCwwog29VYycLCAiYmJpBMJtHU1ASz2bzq914Nh8OB3t5eKpiEEK6+vh5WqxVtbW0IBoP8hGzxCFu2gwrcXjACQHt7+5L9rsv1y/p8vqKkIwC3J36lUinYbLa8FqKsT7ZS47qpFlcOLWBJUdXV1cFiscBsNmN+fp4vDDNH2NbV1WFwcBCBQAA//elP89q5ZP2y9fX1aGlp4VOwAoEA/H7/mvplWauCKIp5xcWwtod4PA6tVgubzVbWW7oulws6nY4XTI/Hwz88CCFEJpPxiMJUKsVPxYLBIB9hq1KpMDk5iWPHjuHRRx/FN7/5zbwWjKxfVqfTIZFI8BMyr9d7x25vIcf46XQ6q50snxa0zHHdW7ZsKfu4bqrFlUXjgUhJsGgsm82Ge+65B52dnbzd4NVXX4XL5eLDBmKxWMGvrdVq0dbWhq1bt6K1tRVyuRwzMzO4fPkyxsbGEAgEcl7UyjQ/P49r165Bo9HkFRfD2h4ikQjq6+vR2dlZ1sWr2+2Gx+NBb28vJEmCx+PBqVOnqGASQpYkCAKMRiO2bduGHTt2wGKxQK1W48qVK3juuefg9Xr5kIOl8rCXo1Qq0dTUhM7OTnR0dGDTpk2IRqOYnJzEpUuXcP36dYTD4SXTXjJltpOZzWbeTraczBYui8WS18lZMVEtrjzKgV2Gx+OB0+nEyMgI3G43nn76aezYsaMkRwVDQ0MQRRFutxtarbaoIeDrySuvvIITJ06gtbUVzz//PG8dUKvV0Ov10Gq1qz6CisfjvMUgM182V+8XG5qgVCr5pYWVlCPrNRc2nlOSpKzvi6JIIzvvRDmwVYRqcXH5/X7s2bMHU1NTeOSRR3DgwAHIZDK+OVBookymdDqN+fl5frE3M182V7/s9PQ0fD4fjEYjmpubV3yPcmW95kK1OG80yKDa1dJEj69//evw+/0YHh6G0Wjk/bKJRALA7V6rtQZ0Z/bLJpPJO/plFxYWcPXqVQDIe2jCjRs34PV6IQgC7HZ7yeKySFHQApYUrFZq8V//+lfs378fe/bswbFjxxAMBuHz+TA3N8d3SwtNlFlKKpXi/bLz8/MA7pwm5vf7MTU1Bb1ej9bW1ryyXsfGxhCNRiGKIrq7uys6aIEsiwYZVLtgMJh17NDf34+9e/dCkqSqO44YHh4GAH5E1NbWhs2bNyMUCmF2djYroFsul/PdgEIDulfql00mk0ilUnkvXsuV9UoIqZxaqcVf+tKX8MEHH6C5uRkymQx6vR56vX7ViTK5rNQvq1arEY1G+eWwQrNey93CRdYX2oGtMI/Hg127dmWFIrPvnTlzpuZCkUs5wjaZTPILW6zXK5982XA4jPHxcaTTadhsNhiNxlW9PymravtUo1pcYlSLs0WjUb6YzRxhuzhRZrWvzYYwAMg7X3ZqaopvQtjt9rLHZZGC0Q5sNaOJHtkWj7BlOwCLA7pZv2whQw0EQUAkEkEqlYLJZEIqlcrKl9VqtRBFMatfdnHWKy1eCalOVIuzqdXqrEQZdkKWmSij1Wqh1+sLHmogCALm5+ehUChgMpmyTslyTROrdNYrWX9oB3Yd2rVrF4UiL5I5wpalFxR64YBdwDKZTDy6K1e/7IULFxCLxWC321FXV4empqayx2WRNam2/6OoFlcA1eJsi0fYMoUMNWD3D1gLF1uILtUvCwBnz56F3W6HyWSCXC7H1q1byx6XRVaNLnHVkoGBAQpFXkY6nYYkSfD5fAgEAvzCAbukpdfrl7xwwC5gGQwGtLS0LLkQTafTfCfg4MGDCAQCUCgUuP/++/GNb3wDX/va14o6ntDlcuHUqVNVd0FknaAFLFkTqsXLSyQS/IQsHA7z77MRtkslyrALWLFYDDabLefQAdYve+bMGbz22msAAKPRiIcffhhPPPEEtmzZUtTfhWpxydACtlY4HI6sUGSyvGQyyftlQ6EQ/35dXR3fDVAqlbhw4QKSySQMBgM2b96c10WBDz74AO+++y4+/vhjXpyfe+45PPnkk2t+brfbDZfLhfPnzyMYDOKtt95a82uSO9AClqwa1eLCRCKRnIkyer0ejY2NiMVi+OSTT3iGdz67qNFoFCdOnMDZs2dx4cIFADdbG86ePQu9Xr/m56ZaXHK0gK0FLpcLkiTRRI9VWjzClvn973+P48eP4+DBg3j22Wfz2kHNzHrt6OjAhx9+CJfLhQMHDuCuu+4q2jM7HA44nU4qmqVBC1iyKlSLVy+dTvNIrkAgwC/LxmIxvPzyy7h69SreeecddHR0rPhai7NeFQoF3nvvPQSDQTz77LNrGlm7GNXikqFLXNWOTfTYvXs3jxk5depUVQZol0rmCFvWL/vee+/h+PHjUKlU+MIXvoCpqSnodDpoNJqcu7DsSEwQBHR3d0Oj0aCvr492YgipAVSL12bxCFu/3w+v14vXXnsNn3/+ObZv345kMomZmZllE2XYuO5EIgFRFNHe3g6ZTIbDhw+X+Tci6xktYCtMkiQcOnQIkiRhYGCAf18UxZIUTTZlho2+O3r0aFXtLLBjq+npabzxxhsQBAE/+MEP0NHRgbm5OczNzUGhUPBbrpkXDljWKwDKeiWkxlAtLi6WKHP8+HF88skn2LJlC374wx8ikUjwRBkWYyiKIj8dY2NlKeuVrIQWsBVWztFzi6fMHDlyBD/60Y+qsnE9lUrBYrHg8OHDeOyxx5BIJHi/bDgchs/ng8/n41NhVCoVJicnAQA2m62qPkgIISujWlwaarUaPT09+OUvf5l1Qub3+xEOhxEOh3mMoU6nQygUQjAYhFKpRHd3d0FRiaS20AK2hvzpT3+CXq/Ht771LQDAjh074HA4KvxUpdHV1YU//OEP/GulUgmTyQSTycRDtG/cuIFoNIpoNMr/Ocp6JYSUWi3V4meeeQbPPPMM/1qj0aC9vR1WqxWSJGF2dpbHGGYONti6dStlvZJl0QK2hvz85z+HTqfjX58/fx7bt2+v4BNVhlqtxubNm2GxWBAKhTAzMwO/34+GhgaYzeZKPx4hpMpRLb7Z7sVaudgIW6/Xu2LEFiEMLWBrSOY0GY/HgwsXLtT0rUs2CIFCsQkh5US1OJtcLkdzczOam5sr/ShkA6EFbA1ikSEvvfRSTY5IJISQ9YBqMSGrRzmwNezIkSPQ6/V46aWXKv0oNcXj8cDpdGJkZARutxtPP/00duzYQVFdxVVt15apFlcxqsWVQbW45GiQASkOSZKybte7XC489dRTOH36NHp6eir4ZIQUHS1gybpFtZjUiJLW4eINdifrmtvtxs6dO+HxePj32CWCubm5kr73U089VdLXJ4SQjYJqMSHFQQvYGmG1WtHf35/VZ+VyuWC1WtHb21uy93U4HHC5XCV7fUII2UioFhNSHHSJq0aIooj+/n4MDQ3x73k8Hpw+fbpk75mZ60cIIYRqMSHFQj2wpGQcDgd6e3uxa9cujI6OVvpxSG2hHlhCbqFaTCqEemDJxuN2u0t6HEYIIWRlVItJtaIFLCmJTz/9lHINCSGkwqgWk2pFPbCk6JxOJ/r7+yv9GGUzNDTEPyA8Hg+fb04IIZVUS7WY6nDtoR1YUlQejycr37DaDQ4OQhRF9PX1oa+vD1arFYODg5V+LEJIjaulWkx1uDbRJS5SVE6nMyvfMBAI4M0338TRo0fR09NTdb1YO3fuxB//+Ef+QSFJEh588EGcO3euwk9W8+gSF6lptVSLqQ6vWzSJi2xcbrcbe/furcqbrx6PZ8lbvXa7nSbqVB4tYAnJUK21mOrwukYpBGRjcjqd+PWvfw3g5hGP2+2u8BMV13JHdKWeqEMIIfmq5lpMdbh20SUuUjKsH6mUXC4XPvzwQ+zfvx9WqxUOhwM6na7k7wvkDgcXRZFCwwkh60apazHVYVIJtANLNjRJkvDmm29i165dsNvtGB8fL0vRBJDzr35Jkmrm8gQhhFAdJpVAO7Bkwztz5gwAlD3rUKfTrepnhBBSbagOk3KjBSzZ8HQ6XUX+0u7p6eHHVJm3X0VRpIsDhJCaQnWYlBu1EJANz+FwwOl0wuFwlD37b9++fRgZGeFfj4yMYN++fWV9BkIIqTSqw6TcVorRImRds9vtnQBujI6OBm59/W0A/zI6OvpvZXyG/wDwOYBNALpGR0e/X673JoSQSqM6TCqBFrCkqtwqpJ8BMLBiSgghpHyoDpNyoB5YsmHZ7XY9AD9u/rX9+a1v37j1350A/q8iD0YIITWC6jCpFOqBJRvdjzOKJnCzYGJ0dJSKJiGElAfVYVJ2tIAlG9atoynfom8fBvDjCjwOIYTUHKrDpFKoB5ZseLea9wMAugD4RkdHqXASQkgZUR0m5UYLWEIIIYQQsqH8Pz5qWOUwXljIAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "# plt.xkcd()\n",
    "fig = plt.figure(figsize=(10, 5))\n",
    "ax = fig.add_subplot(121, projection='3d')\n",
    "ax.plot_surface(x, y, f)\n",
    "ax.set_title('Original function')\n",
    "ax = fig.add_subplot(122, projection='3d')\n",
    "ax.plot_surface(x, y, fappr - f)\n",
    "ax.set_title('Approximation error with rank=%d, err=%3.1e' % (r, er))\n",
    "fig.subplots_adjust()\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сингулярные числа случайной гауссовой матрицы \n",
    "\n",
    "**Q**: Как будут убывать сингулярные числа случайной матрицы?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Singular value index, $i$')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "n = 1000\n",
    "a = np.random.randn(n, n)\n",
    "u, s, v = np.linalg.svd(a)\n",
    "plt.semilogy(s/s[0])\n",
    "plt.ylabel(r\"$\\sigma_i / \\sigma_0$\", fontsize=24)\n",
    "plt.xlabel(r\"Singular value index, $i$\", fontsize=24)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Линейный факторный анализ & малый ранг\n",
    "\n",
    "Рассмотрим линейную модель \n",
    "\n",
    "$$y = Ax, $$ \n",
    "\n",
    "где $y$ вектор размерности $n$, а $x$ вектор размерности $r$.  \n",
    "\n",
    "Данные представлены как набор объектов  \n",
    "\n",
    "$$y_1, \\ldots, y_T,$$\n",
    "\n",
    "но мы не знаем матрицу $A$.\n",
    "Факторная модель можем быть записана как  \n",
    "\n",
    "$$\n",
    "  Y = AX,\n",
    "$$\n",
    "\n",
    "где $Y$ размерности $n \\times T$, $A$ – $n \\times r$ и $X$ – $r \\times T$. \n",
    "\n",
    "Это в точности модель ранга $r$: она означает, что векторы лежат в подпространстве малой размерности!  \n",
    "Мы также можем использовать SVD для восстановления этого подпространства."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Приложения SVD\n",
    "\n",
    "SVD имеет много разных названий, например метод главных компонент (PCA), Proper Orthogonal Decomposition, Empirical Orthogonal Functions, etc\n",
    "\n",
    "Рассмотрим следующие прикладные задачи:\n",
    "\n",
    "1. Латентный семантический анализ\n",
    "2. Коллаборативная фильтрация\n",
    "3. Сжатие данных"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Латентный семантический анализ\n",
    "\n",
    "Одно из самых известных приложений – это поиск с помощью латентного семантического представления, см например\n",
    "[Deerwester, Scott C., et al. \"Indexing by latent semantic analysis.\" (1990)](http://lsa.colorado.edu/papers/JASIS.lsi.90.pdf)  \n",
    "\n",
    "- Постановка задачи: дан набор текстовых документов $D_1, \\ldots, D_N.$ Мы хотим решить задачу поиска, то есть по данному запросу в виде последовательности слов определить наиболее близкие документы.  \n",
    "- Обработка данных устроена следующим образом: для каждого документа создаётся список слов, которые в него входят, и вычисляется частота каждого слова. Такая модель называется **мешок слов** (то есть документ – это неупорядоченный набор слов)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Матрица слов-документов\n",
    "\n",
    "Построим матрицу $A$ слов-документов. Количество строк в ней равно числу слов в словаре. Количество столбцов равно числу документов. Элемент $A_{ij}$ – это частота встречаемости $i$-ого слова в $j$-ом документе.\n",
    "\n",
    "Поиск выполняется умножением матрицы слов-документов на поисковый вектор, то есть на вектор, полученный из поискового запроса."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Проблема**: документ будет выдан только если будет точное совпадение слов в нём и в поисковом запросе. Однако нам бы хотелось, чтобы на запрос \"Samuel Clemens\", мы получили результаты такие же, как и на запрос \"Mark Twain\". Но точных совпадений может и не быть!\n",
    "\n",
    "**Q**: Как может помочь SVD?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Идея латентного семантического представления\n",
    "\n",
    "Вычислим **малоранговое приближение** $A_r$ матрицы слов-документов $A$.  \n",
    "\n",
    "$$A \\approx A_r,$$\n",
    "\n",
    "и **не будем следить за точностью аппроксимации**.  \n",
    "\n",
    "Матрица $A_r$ может быть далее использована для обработки запросов.  \n",
    "\n",
    "Спроецируем документы на малоразмерное подпространство, и для данного запроса $q$ проекция будет равна \n",
    "\n",
    "$$q_r = \\Sigma^{-1}_r U^{\\top}_r q$$\n",
    "\n",
    "Теперь мы сможем вычислить похожесть $q_r$ и других спроецированных документов\n",
    "\n",
    "$$\\widehat{d}_i = \\Sigma^{-1}_r U^{\\top}_r d_i,$$\n",
    "\n",
    "через косинус угла между вектором запроса и спроецированным вектором документа.\n",
    "\n",
    "Демо будет на семинаре на следующей неделе."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Коллаборативная фильтрация\n",
    "\n",
    "- Другое важное и похожее приложение появилось из **рекомендательных систем**.  \n",
    "\n",
    "- Пусть имеется матрица пользователей-продуктов: каждый пользователь ставит рейтинг некоторым продуктам.\n",
    "\n",
    "- Тогда эта матрица имеет размер **число пользователей $\\times$ число продуктов**. Задача – рекомендовать пользователям другие продукты, которые он мог бы купить.\n",
    "\n",
    "- Схема остаётся прежней: вычисляем SVD, и получаем рекомендации для каждого пользователя как столбец приближённой матрицы.\n",
    "\n",
    "Подробнее на семинаре на следующей неделе :)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сжатие плотных матриц\n",
    "\n",
    "Для хранения плотных матриц размера $N \\times N$ обычно необходимо $N^2$ памяти. Аппроксимация ранга $r$ позволяет сократить требуемую память до $\\mathcal{O}(Nr)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Singular value index, $i$')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "# plt.xkcd()\n",
    "n = 256\n",
    "a = [[1.0/(i + j + 0.5) for i in range(n)] for j in range(n)]\n",
    "a = np.array(a)\n",
    "u, s, v = np.linalg.svd(a)\n",
    "plt.semilogy(s/s[0])\n",
    "plt.ylabel(r\"$\\sigma_i / \\sigma_0$\", fontsize=24)\n",
    "plt.xlabel(r\"Singular value index, $i$\", fontsize=24)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Более реалистичный пример\n",
    "\n",
    "Рассмотрим задачу решение линейной системы с матрицей $A$, элементы которой\n",
    "\n",
    "$$A_{ij} = \\frac{1}{i - j + \\frac{1}{2}},$$\n",
    "\n",
    "что соответствует интегральному уравнению\n",
    "\n",
    "$$\n",
    "    \\int \\frac{q(y)dy}{x - y } = f(x).\n",
    "$$\n",
    "\n",
    "На практике, уравнение\n",
    "\n",
    "$$\n",
    "    \\int_{\\Omega} \\frac{q(y)dy}{\\Vert x - y\\Vert } = f(x),\n",
    "$$\n",
    "\n",
    "решается для $\\Omega$ – поверхность в 3D. Это уравнение встречается в задачах моделирования интегральных схем. Посмотрим, что происходит с сингулярными числами."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.884981308350689e-15"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "# plt.xkcd()\n",
    "n = 256\n",
    "a = [[1.0/(i - j + 0.5) for i in range(n)] for j in range(n)]\n",
    "a = np.array(a)\n",
    "u, s, v = np.linalg.svd(a)\n",
    "plt.plot(s / s[0])\n",
    "plt.ylabel(r\"$\\sigma_i / \\sigma_0$\", fontsize=24)\n",
    "plt.xlabel(r\"Singular value index, $i$\", fontsize=24)\n",
    "plt.tight_layout()\n",
    "s[0] - np.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Что делать?\n",
    "\n",
    "Идея: разбить матрицу на блоки\n",
    "\n",
    "$$\n",
    "    A = \\begin{bmatrix}\n",
    "    A_{11} & A_{12} \\\\\n",
    "    A_{21} & A_{22}\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "и оказывается, что блоки $A_{12}$ и $A_{21}$ имеют малый ранг!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Singular value index, $i$')"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a12 = a[:128, 128:]\n",
    "u, s, v = np.linalg.svd(a12)\n",
    "plt.semilogy(s / s[0])\n",
    "plt.ylabel(r\"$\\sigma_i / \\sigma_0$\", fontsize=24)\n",
    "plt.xlabel(r\"Singular value index, $i$\", fontsize=24)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Блочно-малоранговые матрицы\n",
    "\n",
    "- Такие матрицы чаще всего встречаются при решении линейных систем, возникающих в процессе решения уравнений в частных производных \n",
    "\n",
    "- Другие названия: иерархические, мозаично-скелетные"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Резюме первой части\n",
    "\n",
    "- Ранг матрицы\n",
    "- Скелетное разложение и представление матрицы ранга $r$\n",
    "- SVD и теорема Eckart'a-Young'a\n",
    "- Приложения SVD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## План второй части лекции\n",
    "\n",
    "- Линейные системы\n",
    "- Обратная матрица\n",
    "- Число обусловленности\n",
    "- Метод Гаусса (Gaussian elimination)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Линейные системы\n",
    "\n",
    "$$ Ax = f, $$\n",
    "\n",
    "где матрица $A$ и вектор $f$ известны.\n",
    "\n",
    "Задача решения системы линейных уравнений – одна из основных задач вычислительной линейной алгебры.\n",
    "\n",
    "Она возникает при решении следующих задач:\n",
    "\n",
    "- задача линейной регрессии\n",
    "- решение уравнений в частных производных и интегральных уравнений\n",
    "- задачи нелинейной регрессии\n",
    "- задачи оптимизации (методы Ньютона-Рафсона и Гаусса-Ньютона, условия ККТ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Пере- и недоопределённые линейные системы\n",
    "\n",
    "Если система $Au = f$ имеет\n",
    "- больше уравнений, чем неизвестных, она называется **переопределённой** (в общем случае не имеет решений)\n",
    "\n",
    "- меньше уравнений, чем неизвестных, она называется **недоопределённой** (решение неединственно, нужны дополнительные предположения чтобы гарантировать единственность решения)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Существование решений \n",
    "\n",
    "Решение системы линейных уравнений с квадратной матрицей $A$\n",
    "\n",
    "$$A u = f$$\n",
    "\n",
    "существует тогда и только тогда, когда \n",
    "* $\\det A \\ne 0$\n",
    "\n",
    "или\n",
    "\n",
    "* матрица $A$ имеет полный ранг."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Шкала размерностей линейных систем \n",
    "\n",
    "В различных приложениях размерности линейных систем могут быть различны \n",
    "\n",
    "- Малая: $n \\leq 10^4$ (вся матрица помещается в память, **плотные матрицы**)\n",
    "- Средняя: $n = 10^4 - 10^6$ (обычно **разреженные** или **структурированные** матрицы)\n",
    "- Большая: $n = 10^8 - 10^9$ (обычно **разреженные** матрицы и параллельные вычисления)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Линейные системы могут быть большими\n",
    "\n",
    "Возьмём непрерывную задачу, дискретизуем её на сетке из $N$ элементов и получим линейную систему с матрицей $N\\times N$.  \n",
    "Пример сетки для самолёта Airbus A319\n",
    "(подробнее см. [GMSH website](http://geuz.org/gmsh/)).  \n",
    "<img src=\"./a319_4.png\" width=50%>\n",
    "\n",
    "Основная сложность в том, что такие системы очень большие: миллионы и миллиарды неизвестных!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Линейные системы могут быть структурированы\n",
    "\n",
    "- Хранение $N^2$ элементов матрицы невозможно уже для $N = 100000$.  \n",
    "\n",
    "**Q:** как работать с такими матрицами?  \n",
    "\n",
    "**A:** к счастью, такие матрицы чаще всего являются **структурированными** и требуют хранения $\\mathcal{O}(N)$ элементов.\n",
    "\n",
    "- Наиболее растространённый тип структурированных матриц – это разреженные матрицы: такие матрицы имеют только $\\mathcal{O}(N)$ не нулевых элементов!  \n",
    "\n",
    "- Пример (одна из самых известных матриц для $n = 5$):\n",
    "\n",
    "$$\n",
    "  \\begin{pmatrix}\n",
    "  2 & -1 & 0 & 0 & 0 \\\\\n",
    "  -1 & 2 & -1 & 0 & 0 \\\\\n",
    "  0 & -1 & 2 & -1 & 0 \\\\\n",
    "  0 & 0 &-1& 2 & -1  \\\\\n",
    "  0 & 0 & 0 & -1 & 2 \\\\\n",
    "  \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "- По крайней мере можно хранить такие матрицы\n",
    "- Также можно умножать такие матрицы на вектор быстро\n",
    "- Но как решать линейные системы с такими матрицами?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Основные вопросы о линейных системах\n",
    "\n",
    "1. Какую точность мы можем получить от решения (из-за ошибок округления)?\n",
    "2. Как мы вычислим решение? (LU разложение, метод Гаусса)\n",
    "3. Какая сложность решения системы линейных уравнений?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Как решать линейные системы?\n",
    "\n",
    "**Важно**: забудьте о детерминантах и правиле Крамера (хотя они полезны для матриц $2 \\times 2$)!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Как решать линейные системы?\n",
    "\n",
    "Основной инструмент – исключение переменных. \n",
    "\n",
    "\\begin{align*}\n",
    "    &2 y + 3 x = 5 \\quad&\\longrightarrow \\quad &y = 5/2 -  3/2 x \\\\\n",
    "    &2 x + 3z = 5 \\quad&\\longrightarrow\\quad &z = 5/3 - 2/3 x\\\\\n",
    "    &x + y = 2 \\quad&\\longrightarrow\\quad  & 5/2 + 5/3 - (3/2 + 2/3) x = 2,\\\\\n",
    "\\end{align*}\n",
    "\n",
    "и так вы можете найти $x$ (и все остальные неизвестные).  \n",
    "\n",
    "Этот процесс называется **методов Гаусса** и является одним из самых часто используемых алгоритмов. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Метод Гаусса\n",
    "\n",
    "Метод Гаусса состоит из двух этапов:\n",
    "1. Проход вперёд\n",
    "2. Проход назад"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Проход вперёд\n",
    "\n",
    "- Исключим $x_1$:\n",
    "\n",
    "$$\n",
    "   x_1 = f_1 - (a_{12} x_2 + \\ldots + a_{1n} x_n)/a_{11},\n",
    "$$\n",
    "\n",
    "и подставим в уравнения $2, \\ldots, n$. \n",
    "\n",
    "- Затем исключим $x_2$ и подставим в остальные уравнение.\n",
    "\n",
    "- Важно, что ведущий элемент (pivot), тот на который мы делим, не равен $0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Проход назад\n",
    "\n",
    "Во время прохода назад:\n",
    "- решаем уравнение для $x_n$\n",
    "- подставляем решение в уравнение для $x_{n-1}$ и так далее, пока не вычислим все $x_i, i=1,\\ldots, n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Метод Гаусса и LU разложение\n",
    "\n",
    "Метод Гаусса связан с вычислением одного из самых важных матричных разложений: **LU разложения**.\n",
    "\n",
    "**Определение**: LU разложение матрицы $A$ – это представление\n",
    "\n",
    "$$A =  LU,$$\n",
    "\n",
    "где $L$ – **нижнетреугольная** и $U$ – **верхнетреугольная** матрица.\n",
    "\n",
    "Это разложение **неединственно**, поэтому обычно требуют дополнительно, что на диагонали матрицы $L$ стоят 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Основная цель** вычисления LU разложения – это решение системы линейных уравнений, поскольку\n",
    "\n",
    "$$\n",
    "    A^{-1} f = (L U)^{-1} f = U^{-1} L^{-1} f, \n",
    "$$\n",
    "\n",
    "и задача сводится к решению двух линейных систем с верхне- и нижнетреугольных матриц.\n",
    "\n",
    "Проход вперёд выражается в виде\n",
    "$$\n",
    "     L y = f, \n",
    "$$\n",
    "аналогично для прохода назад\n",
    "$$\n",
    "   U x = y.\n",
    "$$\n",
    "\n",
    "Всегда ли существует $LU$ разложение?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сложность метода Гаусса/LU разложения\n",
    "\n",
    "- Каждый шаг исключения занимает $\\mathcal{O}(n^2)$ операций. \n",
    "\n",
    "- Таким образом, сложность алгоритма $\\mathcal{O}(n^3)$.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Существование LU разложения\n",
    "\n",
    "Алгоритм вычисления LU разложения работает,\n",
    "\n",
    "если **мы не делим на $0$** на каждом шаге метода Гаусса.\n",
    "\n",
    "**Q:** Для какого класса матриц это так?\n",
    "\n",
    "**A:** Это так для **строго регулярных матриц**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Строго регулярные матрицы и LU разложение\n",
    "\n",
    "**Определение.** Матрица $A$ называется *строго регулярной*, если все лидирующие главные миноры (подматрицы из первых $k$ строк и $k$ столбцов) не вырождены. \n",
    "\n",
    "В этом случае LU разложение всегда существует. Обратное также верно (проверьте!)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## LU разложение для положительно определённых Эрмитовых матриц (разложение Холецкого)\n",
    "\n",
    "Строго регулярные матрицы имеют LU разложение. \n",
    "\n",
    "Важный класс строго регулярных матриц – это класс **Эрмитовых положительно определённых матриц**\n",
    "\n",
    "**Определение.** Матрица $A$ называется <font color='red'> положительно определённой </font>, если для любого $x: \\Vert x \\Vert \\ne 0$ выполнено\n",
    "\n",
    "$$\n",
    "(x, Ax) > 0.\n",
    "$$\n",
    "- если это выполнено для $x \\in \\mathbb{C}^n$, тогда матрица $A$ эрмитова\n",
    "- если это выполнено для $x \\in \\mathbb{R}^n$, тогда матрица $A$ может быть несимметрична"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Утверждение:** Эрмитова положительно определённая матрица $A$ строго регулярна и имеет разложение Холецкого вида\n",
    "\n",
    "$$A = RR^*,$$\n",
    "\n",
    "где $R$ нижнетреугольная матрица.\n",
    "\n",
    "Часто матрица $R$ называется \"квадратным корнем\" матрицы $A$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вычисление LU разложения\n",
    "\n",
    "Во многих случаях достаточно один раз вычислить LU разложение!\n",
    "\n",
    "Если такое разложение найдено (что требует $\\mathcal{O}(n^3)$ операций), тогда решение линейной системы сводится к решению линейных систем с матрицами $L$ и $U$, которые требуют $\\mathcal{O}(n^2)$ операций.\n",
    "\n",
    "**Упражнение:** Решение линейной системы с треугольной матрицей вычисляется быстро. Как вычислить $L$ и $U$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Когда алгоритм вычисления LU разложения не работает\n",
    "\n",
    "- Что случится, если матрица не является строго регулярной (или ведущий элемент в методе Гаусса очень мал)?   \n",
    "\n",
    "- Классический пример матрицы $2 \\times 2$ с \"плохим\" LU разложением:\n",
    "\n",
    "$$\n",
    "    A = \\begin{pmatrix}\n",
    "    \\varepsilon & 1 \\\\\n",
    "    1 & 1 \n",
    "    \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "- Если $\\varepsilon$ достаточно мал, мы можем встретиться с неустойчивостью. В то время как вычисление разложения Холецкого всегда устойчиво.\n",
    "\n",
    "Проверим это численно..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L * U - A:\n",
      " [[ 0.00000000e+00  0.00000000e+00]\n",
      " [-1.11022302e-16 -1.00000000e+00]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([[1.e+00, 0.e+00],\n",
       "       [1.e+18, 1.e+00]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "eps = 1e-18#1.12e-16\n",
    "a = [[eps, 1],[1.0,  1]]\n",
    "a = np.array(a)\n",
    "a0 = a.copy()\n",
    "n = a.shape[0]\n",
    "L = np.zeros((n, n))\n",
    "U = np.zeros((n, n))\n",
    "for k in range(n): #Eliminate one row   \n",
    "    L[k, k] = 1\n",
    "    for i in range(k+1, n):\n",
    "        L[i, k] = a[i, k] / a[k, k]\n",
    "        for j in range(k+1, n):\n",
    "            a[i, j] = a[i, j] - L[i, k] * a[k, j]\n",
    "    for j in range(k, n):\n",
    "        U[k, j] = a[k, j]\n",
    "\n",
    "print('L * U - A:\\n', np.dot(L, U) - a0)\n",
    "L"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Выбор ведущего элемента (pivoting)\n",
    "\n",
    "Мы можем переставить строки и столбцы матрицы $A$ так, чтобы элемент $A_{kk}$, на который мы делим, был максимальным.  \n",
    "\n",
    "Простейшая, но эффективная стратегия – это выбор максимального элемента в строке: на каждом шаге выбираем элемент, который максимален по модулю и перемещаем его на диагональ.\n",
    "\n",
    "Это даёт следующее разложение \n",
    "\n",
    "$$A = P L U,$$\n",
    "\n",
    "где $P$ – это **матрица перестановки**.\n",
    "\n",
    "\n",
    "- Почему это хорошая стратегия?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Устойчивость линейных систем\n",
    "\n",
    "- Существует фундаментальная проблема с решением систем линейных уравнений, которая не зависит от используемого алгоритма.\n",
    "\n",
    "- Она проявляется, когда элементы матрицы представляются как числа с плавающей точкой или имеется некоторый шум в измерениях.\n",
    "\n",
    "Проиллюстрируем эту проблему на следующем примере."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.095164779214391\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x12297b9e8>]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import seaborn as sns\n",
    "sns.set_style(\"whitegrid\")\n",
    "sns.set_context(\"talk\")\n",
    "n = 15\n",
    "a = [[1.0/(i + j + 1) for i in range(n)] for j in range(n)]\n",
    "a = np.array(a)\n",
    "rhs = np.random.randn(n) #Right-hand side\n",
    "x = np.linalg.solve(a, rhs) #This function computes LU-factorization and solves linear system\n",
    "\n",
    "#And check if everything is fine\n",
    "er = np.linalg.norm(a.dot(x) - rhs) / np.linalg.norm(rhs)\n",
    "print(er)\n",
    "plt.plot(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Линейные системы и обратная матрица\n",
    "\n",
    "- В чём проблема в предыдущем примере? \n",
    "\n",
    "- Почему ошибка растёт так быстро?  \n",
    "\n",
    "- И мы приходим к одному из главных понятий вычислительной линейной алгебры: числу обусловленности матрицы.  \n",
    "\n",
    "Но перед этим нам нужно определить **обратную матрицу**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Обратная матрица: определение\n",
    "\n",
    "Матрица, обратная к матрице $A$, это такая матрица $X$ что  \n",
    "\n",
    "$$\n",
    "   AX = XA = I, \n",
    "$$\n",
    "\n",
    "где $I$ – единичная матрица. Обратная матрица обозначается как $A^{-1}$.\n",
    "\n",
    "Вычисление обратной матрицы связано с решением линейной системы. В самом деле, $i$-ый столбец произведения даёт\n",
    "\n",
    "$$\n",
    "A x_i = e_i,\n",
    "$$\n",
    "\n",
    "где $e_i$ – $i$-ый столбец единичной матрицы. \n",
    "Таким образом, мы можем использовать метод Гаусса, чтобы решить эту систему.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Обратная матрица и линейные системы\n",
    "\n",
    "Если у нас есть обратная матрица $A^{-1}$, тогда решение линейной системы\n",
    "\n",
    "$$Ax = f$$\n",
    "\n",
    "выражается как $x = A^{-1} f$.  \n",
    "\n",
    "В самом деле,  \n",
    "\n",
    "$$   \n",
    "    A(A^{-1} f) = (AA^{-1})f = I f = f.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ряд Неймана\n",
    "\n",
    "Для того чтобы выяснить, почему в решении была такая большая ошибка нам потребуется важный вспомогательный результат.  \n",
    "\n",
    "**Ряд Неймана**:  \n",
    "\n",
    "Если матрица $F$ такая что $\\Vert F \\Vert < 1$, тогда матрица $(I - F)$ обратима и\n",
    "\n",
    "$$(I - F)^{-1} = I + F + F^2 + F^3 + \\ldots = \\sum_{k=0}^{\\infty} F^k.$$\n",
    "\n",
    "Заметим, что это матричная версия выражения для суммы геометрической прогрессии. \n",
    "\n",
    "**Q**: какая норма тут используется? Какая норма \"самая лучшая\" в данном случае?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Доказательство\n",
    "\n",
    "Сначала докажем, что ряд $\\sum_{k=0}^{\\infty} F^k$ сходится.  \n",
    "\n",
    "Как и в скалярном случае выполнено\n",
    "$$\n",
    "   (I - F) \\sum_{k=0}^N F^k = (I - F^{N+1}) \\rightarrow I, \\quad N \\to +\\infty\n",
    "$$\n",
    "\n",
    "Действительно, \n",
    "$$\n",
    "\\| (I - F^{N+1}) - I\\| = \\|F^{N+1}\\| \\leqslant \\|F\\|^{N+1} \\to 0, \\quad N\\to +\\infty.\n",
    "$$\n",
    "\n",
    "Также можем оценить **норму обратной матрицы**:\n",
    "$$\n",
    "  \\left\\Vert \\sum_{k=0}^N F^k \\right\\Vert \\leq \\sum_{k=0}^N \\Vert F \\Vert^k \\Vert I \\Vert \\leq \\frac{\\Vert I \\Vert}{1 - \\Vert F \\Vert} \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Малое возмущение обратной матрицы\n",
    "\n",
    "Используя этот результат, мы можем оценить как возмущение в элементах матрицы влияет на возмущение в элементах обратной матрицы. Предположим, что возмущение $E$ мало в том смысле, что $\\Vert A^{-1} E \\Vert < 1$. Тогда\n",
    "\n",
    "$$(A + E)^{-1} = \\sum_{k=0}^{\\infty} (-A^{-1} E)^k A^{-1}$$\n",
    "\n",
    "и более того, \n",
    "\n",
    "$$\n",
    "  \\frac{\\Vert (A + E)^{-1} - A^{-1} \\Vert}{\\Vert A^{-1} \\Vert} \\leq \\frac{\\Vert A^{-1} \\Vert \\Vert E \\Vert \\Vert I \\Vert}{1 - \\Vert A^{-1} E \\Vert}.\n",
    "$$\n",
    "\n",
    "Видно, что норма обратной матрицы входит в оценку."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Число обусловленности линейной системы\n",
    "\n",
    "Рассмотрим **возмущённую** линейную систему:\n",
    "\n",
    "$$\n",
    "   (A + \\Delta A) \\widehat{x} = f + \\Delta f.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Оценки!\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "\\widehat{x} - x &= (A + \\Delta A)^{-1} (f + \\Delta f) - A^{-1} f =\\\\ \n",
    "&= \\left((A + \\Delta A)^{-1} - A^{-1}\\right)f + (A + \\Delta A)^{-1} \\Delta f = \\\\\n",
    "&= \\Big[\\sum_{k=0}^{\\infty} (-A^{-1} \\Delta A)^k\\Big] A^{-1} f + \\Big[\\sum_{k=0}^{\\infty} (A^{-1} \\Delta A)^k \\Big] A^{-1} \\Delta f,\n",
    "\\end{split}\n",
    "$$  \n",
    "поэтому\n",
    "$$\n",
    "\\begin{split}\n",
    "\\frac{\\Vert \\widehat{x} - x \\Vert}{\\Vert x \\Vert} \\leq \n",
    "&\\frac{\\Vert A \\Vert \\Vert A^{-1} \\Vert}{1 - \\|A^{-1}\\Delta A\\|} \\Big(\\frac{\\Vert\\Delta A\\Vert}{\\Vert A \\Vert} + \\frac{\\Vert \\Delta f \\Vert}{ \\Vert f \\Vert}\\Big) \\leq \\\\\n",
    "\\leq\n",
    "&\\frac{\\Vert A \\Vert \\Vert A^{-1} \\Vert}{1 - \\|A\\|\\|A^{-1}\\|\\frac{\\|\\Delta A\\|}{\\|A\\|}} \\Big(\\frac{\\Vert\\Delta A\\Vert}{\\Vert A \\Vert} + \\frac{\\Vert \\Delta f \\Vert}{ \\Vert f \\Vert}\\Big) \\equiv \\\\\n",
    "\\equiv &\\frac{\\mathrm{cond}(A)}{1 - \\mathrm{cond}(A)\\frac{\\|\\Delta A\\|}{\\|A\\|}} \\Big(\\frac{\\Vert\\Delta A\\Vert}{\\Vert A \\Vert} + \\frac{\\Vert \\Delta f \\Vert}{ \\Vert f \\Vert}\\Big)\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "Главную роль играет **число обусловленности** матрицы $A$: $\\mathrm{cond}(A) = \\Vert A \\Vert \\Vert A^{-1} \\Vert$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Число обусловленности\n",
    "\n",
    "- Чем больше число обусловленности, тем меньше цифр в записи числа мы можем правильно восстановить. \n",
    "- Число обусловленности разное для разных норм.\n",
    "- Для спектральной нормы выполнено (проверьте!) $\\mathrm{cond}_2 (A) = \\|A\\|_2 \\|A^{-1}\\|_2 = \\frac{\\sigma_{\\max}}{\\sigma_{\\min}}$\n",
    "- Заметим, что если $\\Delta A = 0$, тогда\n",
    "\n",
    "$$\n",
    "\\frac{\\Vert \\widehat{x} - x \\Vert}{\\Vert x \\Vert} \\leq \\mathrm{cond}(A) \\frac{\\|\\Delta f\\|}{\\|f\\|}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## И снова матрица Гильберта \n",
    "\n",
    "- Проверим насколько оценка точная в обоих случаях: единичная правая часть и случайная правая часть (вспомните пример с прошлой лекции!)\n",
    "- Результаты существенно отличаются!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error: 2.217472233227586e-06 Condition number: 4.492167979846466e+20\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "\n",
    "n = 1000\n",
    "a = [[1.0/(i + j + 1) for i in range(n)] for j in range(n)]\n",
    "a = np.array(a)\n",
    "rhs = np.ones(n) #Right-hand side\n",
    "f = np.linalg.solve(a, rhs)\n",
    "\n",
    "#And check if everything is fine\n",
    "er = np.linalg.norm(a.dot(f) - rhs) / np.linalg.norm(rhs)\n",
    "cn = np.linalg.cond(a, 2)\n",
    "print('Error:', er, 'Condition number:', cn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error: 9.698022727227913 Condition number: 4.073996146476839e+19\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x122d00048>]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "n = 100\n",
    "a = [[1.0/(i + j + 1) for i in range(n)] for j in range(n)]\n",
    "a = np.array(a)\n",
    "rhs = np.random.randn(n) #Right-hand side\n",
    "f = np.linalg.solve(a, rhs)\n",
    "\n",
    "#And check if everything is fine\n",
    "er = np.linalg.norm(a.dot(f) - rhs) / np.linalg.norm(rhs)\n",
    "cn = np.linalg.cond(a)\n",
    "print('Error:', er, 'Condition number:', cn)\n",
    "\n",
    "u, s, v = np.linalg.svd(a)\n",
    "rhs = np.random.randn(n)\n",
    "plt.plot(u.T.dot(rhs))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Как это объяснить?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Переопределённые линейные системы\n",
    "\n",
    "- Рассмотрим переопределённые линейные системы, в которых число уравнений больше, чем число неизвестных.\n",
    "- Простейший пример: аппроксимация точек на плоскости с помощью линейной модели\n",
    "\n",
    "Стандартный способ минимизации невязки (**линейная задача наименьших квадратов**)\n",
    "\n",
    "$$\\Vert A x - b \\Vert_2 \\rightarrow \\min$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Переопределённая система и матрица Грама\n",
    "\n",
    "Условие оптимальности $0\\equiv \\nabla \\left(\\|Ax-b\\|_2^2\\right)$, где $\\nabla$ обозначает градиент. Поэтому,\n",
    "\n",
    "$$\n",
    "0 \\equiv \\nabla \\left(\\|Ax-b\\|_2^2\\right) = 2(A^*A x - A^*b) = 0.\n",
    "$$\n",
    "\n",
    "Таким образом,\n",
    "\n",
    "$$\n",
    "A^* A x = A^* b\n",
    "$$\n",
    "\n",
    "Матрица $A^* A$ называется **матрицей Грама**, а система называется **нормальным уравнением**. \n",
    "\n",
    "- Число обусловленности матрицы $A^* A$ равно квадрату числа обусловленности матрицы $A$ (проверьте!).\n",
    "- Поэтому решать нормальное уравнение в таком виде – не самая хорошая идея!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Псевдообратная матрица\n",
    "\n",
    "Матрица $A^* A$ может быть вырождена в общем случае (почему?).\n",
    "Поэтому, необходимо ввести понятие псевдообратной матрицы  $A^{\\dagger}$ такой что <br>\n",
    "решение линейной задачи наименьших квадратов можно было записать в виде\n",
    "\n",
    "$$x = A^{\\dagger} b.$$\n",
    "\n",
    "Матрица $$A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^* A)^{-1} A^*$$ называется псевдообратной матрицей Мура-Пенроуза для матрицы $A$.\n",
    "\n",
    "* Если матрица $A$ имеет полный ранг, тогда $A^* A$ невырождена, и мы получим $A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^* A)^{-1} A^* = (A^* A)^{-1} A^*$.\n",
    "\n",
    "* Если матрица $A$ квадратная и невырожденная, мы получим $A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^* A)^{-1} A^* = (A^* A)^{-1} A^* = A^{-1} A^{-*} A^* = A^{-1}$ – обычная обратная матрица для $A$\n",
    "\n",
    "* Если $A$ имеет линейно зависимые столбцы, тогда $A^\\dagger b$ даёт решение минимальной евклидовой нормы. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вычисление псевдообратной матрицы с помощью SVD\n",
    "\n",
    "Пусть $A = U \\Sigma V^*$ SVD для матрицы $A$. Тогда,\n",
    "\n",
    "$$A^{\\dagger} = V \\Sigma^{\\dagger} U^*,$$\n",
    "\n",
    "где $\\Sigma^{\\dagger}$ состоит из обращённых ненулевых сингулярных чисел матрицы $A$. Действительно,\n",
    "\n",
    "$$A^{\\dagger} = \\lim_{\\alpha \\rightarrow 0}(\\alpha I + A^* A)^{-1} A^* = \\lim_{\\alpha \\rightarrow 0}( \\alpha VV^* + V \\Sigma^2 V^*)^{-1} V \\Sigma U^* = \\lim_{\\alpha \\rightarrow 0}( V(\\alpha I + \\Sigma^2) V^*)^{-1} V \\Sigma U^* = V \\lim_{\\alpha \\rightarrow 0}(\\alpha I + \\Sigma^2)^{-1} \\Sigma U^* = V \\Sigma^{\\dagger} U^*,$$\n",
    "\n",
    "* Вы можете проверить, что $\\Sigma^{\\dagger}$ состоит из обращённых ненулевых сингулярных чисел <br>\n",
    "* Если сингулярные числа малы, их можно не обращать. Это даст решение менее чувствительное к шуму в правой части\n",
    "\n",
    "**Q:** что произошло с числом обусловленности?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Стандартный способ решения линейной задачи наименьших квадратов\n",
    "\n",
    "Использование $QR$ разложения.\n",
    "\n",
    "Любая матрица может быть представлена в виде \n",
    "\n",
    "$$\n",
    "A = Q R,\n",
    "$$\n",
    "где $Q$ – унитарная матрица, и $R$ – верхнетреугольная.\n",
    "\n",
    "Тогда, если $A$ имеет полный ранг, тогда\n",
    "\n",
    "$$\n",
    "x = A^{\\dagger}b = (A^*A)^{-1}A^*b = ((QR)^*(QR))^{-1}(QR)^*b = (R^*Q^*QR)^{-1}R^*Q^*b = R^{-1}Q^*b. \n",
    "$$ \n",
    "\n",
    "Таким образом, задача поиска оптимального $x$ эквивалентна решению следующей квадратной системы \n",
    "\n",
    "$$\n",
    "Rx = Q^* b.\n",
    "$$\n",
    "\n",
    "Так как $R$ верхнетреугольная, решение этой системы требует $\\mathcal{O}(n^2)$ операций. Также этот способ более устойчив, чем использование псевдообратной матрицы напрямую."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Пример линейной задачи наименьших квадратов\n",
    "\n",
    "Рассмотрим двумерный пример. Пусть дана линейная модель \n",
    "\n",
    "$$y = ax + b$$\n",
    "\n",
    "и зашумлённые данные $(x_1, y_1), \\dots (x_n, y_n)$. Тогда линейная система на коэффициенты будет выглядеть как \n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "a x_1 &+ b &= y_1 \\\\\n",
    "&\\vdots \\\\\n",
    "a x_n &+ b &= y_n \\\\\n",
    "\\end{split}\n",
    "$$\n",
    "или в матричном виде\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "x_1 & 1 \\\\\n",
    "\\vdots & \\vdots \\\\\n",
    "x_n & 1 \\\\\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "a \\\\\n",
    "b\n",
    "\\end{pmatrix} =\n",
    "\\begin{pmatrix}\n",
    "y_1 \\\\\n",
    "\\vdots  \\\\\n",
    "y_n \\\\\n",
    "\\end{pmatrix},\n",
    "$$\n",
    "что является переопределённой системой"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x123548668>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "a_exact = 1.\n",
    "b_exact = 2.\n",
    "\n",
    "n = 10\n",
    "xi = np.arange(n)\n",
    "yi = a_exact * xi + b_exact + 2*np.random.random(n)\n",
    "\n",
    "A = np.array([xi, np.ones(n)])\n",
    "coef = np.linalg.pinv(A).T.dot(yi) # coef is [a, b]\n",
    "\n",
    "plt.plot(xi, yi, 'o', label='$(x_i, y_i)$')\n",
    "plt.plot(xi, coef[0]*xi + coef[1], label='Least Squares')\n",
    "plt.legend(loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Резюме по второй части\n",
    "\n",
    "- Линейные системы можно решать методом Гаусса, сложность – $\\mathcal{O}(n^3)$.\n",
    "- Линейные системы можно решать с помощью LU разложения, сложность – $\\mathcal{O}(n^3)$ для разложения, и  $\\mathcal{O}(n^2)$ для каждой правой части\n",
    "- Линейная заадча наименьших квадратов может быть решена с помощью решения нормального уравнения (плохая идея!)\n",
    "- Линейная заадча наименьших квадратов может быть решена с помощью QR разложения (стандартный подход)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Анонс на следующую неделю\n",
    "\n",
    "- Лекция и семинар\n",
    "- Подготовьте вопросы по этой длинной лекции\n",
    "- Будем разбирать приложения и практические аспекты всего того, что сегодня успели обсудить"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
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}