forked from CalculusWithJulia/CalculusWithJulia.github.io
-
Notifications
You must be signed in to change notification settings - Fork 0
/
lhopitals_rule.html
644 lines (530 loc) · 99.6 KB
/
lhopitals_rule.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link
href="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/css/bootstrap.min.css"
rel="stylesheet">
<style>
.julia {font-family: "Source Code Pro";
color:#0033CC;
}
body { padding-top: 60px; }
h5:before {content:"\2746\ ";}
h6:before {content:"\2742\ ";}
pre {display: block;}
</style>
<script src="http://code.jquery.com/jquery.js"></script>
<script src="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
<!-- .julia:before {content: "julia> "} -->
<style></style>
<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<script>
MathJax.Hub.Config({
tex2jax: {
inlineMath: [ ["\$","\$"], ["\\(","\\)"]]
},
displayAlign: "left",
displayIndent: "5%"
});
</script>
<script type="text/javascript">
$( document ).ready(function() {
$("h1").each(function(index) {
var title = $( this ).text()
$("#page_title").html("<strong>" + title + "</strong>");
document.title = title
});
$( "h2" ).each(function( index ) {
var nm = $( this ).text();
var id = $.trim(nm).replace(/ /g,'');
this.id = id
$("#page_dropdown").append("<li><a href='#" + id + "'>" + nm + "</a></li>");
});
$('[data-toggle="popover"]').popover();
});
</script>
</head>
<body data-spy="scroll" >
<nav class="navbar navbar-default navbar-fixed-top">
<div class="container-fluid">
<!-- Brand and toggle get grouped for better mobile display -->
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#bs-example-navbar-collapse-1">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="../toc.html">Calculus with Julia</a>
</div>
<!-- Collect the nav links, forms, and other content for toggling -->
<div class="collapse navbar-collapse" id="bs-example-navbar-collapse-1">
<ul class="nav navbar-nav">
<li><a href="#" id="page_title"></a></li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">
Jump to... <span class="caret"></span></a>
<ul class="dropdown-menu" role="menu" id="page_dropdown"></ul>
</li>
</ul>
</div><!-- /.navbar-collapse -->
</div><!-- /.container-fluid -->
</nav>
<header>
</header>
<div class="container-fluid">
<div class="span10 offset1">
<h1>L'Hopitals Rule</h1><p>Let's return to limits of the form $f(x)/g(x)$ which have an indeterminate form of $0/0$ if both are evaluated at $c$. The typical example being the limit considered by Euler:</p>$$~
\lim_{x\rightarrow 0} \frac{\sin(x)}{x}.
~$$<p>We know this is $1$ using a bound from geometry, but might also guess this is one, as we know from linearization near $0$ that we have</p>$$~
\sin(x) = x - \sin(\xi)x^2/2, \quad 0 < \xi < x.
~$$<p>This would yield:</p>$$~
\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = \lim_{x\rightarrow 0} \frac{x -\sin(\xi) x^2/2}{x} = \lim_{x\rightarrow 0} 1 + \sin(\xi) \cdot x/2 = 1.
~$$<p>This is because we know $\sin(\xi) x/2$ has a limit of $0$, when $|\xi| \leq |x|$.</p><p>That doesn't look any easier, as we worried about the error term, but if just mentally replaced $\sin(x)$ with $x$ – which it basically is near $0$ – then we can see that the limit should be the same as $x/x$ which we know is $1$ without thinking.</p><p>Basically, we found that in terms of limits, if both $f(x)$ and $g(x)$ are $0$ at $c$, that we <em>might</em> be able to just take this limit: $(f(c) + f'(c) \cdot(x-c)) / (g(c) + g'(c) \cdot (x-c))$ which is just $f'(c)/g'(c)$.</p><p>Wouldn't that be nice? We could find difficult limits just by differentiating the top and the bottom at $c$ (and not use the messy quotient rule).</p><p>Well, in fact that is more or less true, a fact that dates back to <a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule">L'Hopital</a> – who wrote the first textbook on differential calculus – though this result is likely due to one of the Bernoulli brothers.</p><blockquote>
<p><strong>L'Hopital's rule</strong>: Suppose, $f$ and $g$ are differentiable in $(a,b)$ with $a < c < b$. Moreover, suppose $g(x) > 0$ for all $x$ in $(a,b)$ except at $c$. Further suppose $f(c)=g(c) =0$. If $\lim_{x\rightarrow c}f'(x)/g'(x)=L$ (the limit exists), then $L = \lim_{x \rightarrow c}f(x)/g(x)$.</p>
</blockquote><p>That is <em>if</em> the limit of $f(x)/g(x)$ is indeterminate of the form $0/0$, but the limit of $f'(x)/g'(x)$ is known, possibly by simple continuity, then the limit of $f(x)/g(x)$ exists and is equal to that of $f'(x)/g'(x)$.</p><p>To apply this rule to Euler's example, $\sin(x)/x$, we just need to consider that:</p>$$~
1 = \lim_{x \rightarrow 0}\frac{\cos(x)}{1}
~$$<p>This is due to $\cos(x)$ being continuous at $0$, so this limit is just $\cos(0)/1$. (More importantly, the tangent line expansion of $\sin(x)$ at $0$ is $\sin(0) + \cos(0)x$, so that $\cos(0)$ is why this answer is as it is, but we don't need to think in terms of $\cos(0)$, but rather the tangent-line expansion, which is $\sin(x) \approx x$, as $\cos(0)$ appears as the coefficient.</p><h5>Examples</h5><ul>
<li>Consider this limit at $0$: $(a^x - 1)/x$. We have $f(x) =a^x-1$ has $f(0) = 0$, so this limit is indeterminate of the form $0/0$. The derivative of $f(x)$ is $f'(x) = a^x \log(a)$ which has $f'(0) = 1$. The derivative of the bottom is also $1$ at $0$, so we have:</li>
</ul>$$~
1 = \frac{1}{1} = \frac{f'(0)}{g'(0)} = \lim_{x \rightarrow 0}\frac{f'(x)}{g'(x)} = \lim_{x \rightarrow 0}\frac{f(x)}{g(x)}
= \lim_{x \rightarrow 0}\frac{a^x - 1}{x}.
~$$<p>(Why rewrite in the "opposite" direction? Because the theorem's result ($L$ is the limit) is only true if the related limit involving the derivative exists.)</p><ul>
<li>Consider this limit:</li>
</ul>$$~
\lim_{x \rightarrow 0} \frac{e^x - e^{-x}}{x}.
~$$<p>It too is of the indeterminate form $0/0$. The derivative of the top is $e^x + e^{-x}$, which is $2$ when $x=0$, so the ratio of $f'(0)/g'(0)$ is seen to be 2. By L'Hopital's rule, the limit above is $2$.</p><ul>
<li>Sometimes, L'Hopital's rule must be applied twice. Consider this limit:</li>
</ul>$$~
\lim_{x \rightarrow 0} \frac{\cos(x)}{1 - x^2}
~$$<p>By L'Hopital's rule <em>if</em> this following limit exists, the two will be equal:</p>$$~
\lim_{x \rightarrow 0} \frac{-\sin(x)}{-2x}.
~$$<p>But if we didn't guess the answer, we see that this is indeterminate of the form $0/0$. So that limit will exist and be equal to this limit, should it exist:</p>$$~
\lim_{x \rightarrow 0} \frac{-\cos(x)}{-2} = 1/2.
~$$<p>As $L = 1/2$ for this related limit, it must also be the limit of the original problem, by L'Hopital's rule.</p><ul>
<li>Our "intuitive" limits can bump into issues. Take for example the limit of $(\sin(x)-x)/x^2$ as $x$ goes to $0$. Using $\sin(x) \approx x$ makes this look like $0/x^2$ which is still indeterminate. (Because the difference is higher order than $x$.) Using L'Hopitals, says this limit will exist (and be equal) if the following one does:</li>
</ul>$$~
\lim_{x \rightarrow 0} \frac{\cos(x) - 1}{2x}.
~$$<p>This particular limit is indeterminate of the form $0/0$, so we again try L'Hopital's rule and consider</p>$$~
\lim_{x \rightarrow 0} \frac{-\sin(x)}{2} = 0
~$$<p>So as this limit exists, working backwards, the original limit in question will also be $0$.</p><ul>
<li>This example comes from the Wikipedia page. It "proves" a discrete approximation for the second derivative.</li>
</ul><p>Show if $f''(x)$ exists at $c$ and is continuous at $c$, then</p>$$~
f''(c) = \lim_{h \rightarrow 0} \frac{f(c + h) - 2f(c) + f(c-h)}{h^2}.
~$$<p>This will follow from two applications of L'Hopital's rule to the right hand side. The first says, the limit on the right is equal to this limit, should it exist:</p>$$~
\lim_{h \rightarrow 0} \frac{f'(c+h) - 0 - f'(c-h)}{2h}.
~$$<p>We have to be careful, as we differentiate in the $h$ variable, not the $c$ one, so the chain rule brings out the minus sign. But again, as we still have an indeterminate form $0/0$, this limit will equal the following limit should it exist:</p>$$~
\lim_{h \rightarrow 0} \frac{f''(c+h) - 0 - (-f''(c-h))}{2} =
\lim_{c \rightarrow 0}\frac{f''(c+h) + f''(c-h)}{2} = f''(c).
~$$<p>That last equality follows, as it is assumed that $f''(x)$ exists at $c$ and is continuous, that is, $f''(c \pm h) \rightarrow f''(c)$.</p><p>The expression above finds use when second derivatives are numerically approximated. (The middle expression is the basis of the central-finite difference approximation to the derivative.)</p><h2>Why?</h2><p>The proof of L'Hopital's rule takes advantage of Cauchy's <a href="http://en.wikipedia.org/wiki/Mean_value_theorem#Cauchy.27s_mean_value_theorem">generalization</a> of the mean value theorem to two functions. Let $f(x)$ and $g(x)$ be continuous on $[a,b]$ and differentiable on $(a,b)$ so that on $[c,c+x]$ there exists a $\xi$ with $f'(\xi) \cdot (f(x) - f(c)) = g'(c) \cdot (g(x) - g(c))$. In our formulation, both $f(c)$ and $g(c)$ are zero, so we have, provided we know that $g(x)$ is non zero, that $f(x)/g(x) = f'(\xi)/g'(\xi)$ for some $\xi$, $c < \xi < c + x$. That the right hand side has a limit as $x \rightarrow c+$ is true by the assumption that the limit of the derivative's ratio exists. (The $\xi$ part can be removed by considering it as a composition of a function going to $c$.) Thus the right limit of the left hand side is known. Similarly, working with $[c-x, c]$ we can get the left limit is known and is equal to the right.</p><h3>L'Hopital's picture</h3><p><div class="well well-sm"> <figure> <img src="data:image/gif;base64, R0lGODlhFwJ6AfcAAFZKVVhMV1lNWF1TXWVaZGVaZGxga3ZrdXlueHVpc31zff+mAv+pC/+sE/+uGv+wHsyqM9moJvyuIP+yJP+0LP+3M/+4Nv+6PKytU5Swa5a1eqiyY/69RP++SvW9Wf/BTv/DVP/FW/7GYf/JZfHEav/La/XCZ//Oc//PeP/RfOnIcoZ9hYh+h4l/iHazika4uh+83zC60Bq95gG//w6+8QXA/wvC/xPE/xvG/yHH/iTI/yzK/zPM/zzO/yjB50HO/kjP/lPN+UXQ/0vS/1TU/1vW/2nN9GDX/2XZ/2za/3Tc/33e/3DX/HzV5IyDi5CIj5OKkpiPl52VnJiQl6Oco6egpqukq66orrOts7extru2u765vsC6v/XMhP/Tg//WjO/MkfHPlv/ZlP/cnP/eo/baqerYv//gp//irP/ks/7muv/ovsK9wJ3T7oHe/o3b+InU9Lre8ajc84Xg/4ri/5Pk/5vm/6Hn/6To/6zq/6rj+bTs/7zu/7Ph9cfCxsjDx8rGys7KzszextHN0dXS1drW2d3a3ency9bgw//qw//tzPXlyf/v0vLo2P/w1f/y3MTd7dra5t3d6NXV4+Hf4d3j7s3m88Hu/sXw/8vy/9T0/9z2/9/09uTj5OXl7uzq7PTu5f/14//36f/47PXx7ezs8uPq8/Dv8OH2/uX4/+z6/+nz+vLy9vj39//78/b2+fT8/////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACH5BAAAAAAAIf8LTkVUU0NBUEUyLjADAQAAACwAAAAAFwJ6AQAI/gBjCRxIsKDBgwgTKlzIsKHDhxAjSpxIsaLFixgzatzIsaPHjyBDihxJsqTJkyhTqlzJsqXLlzBjypxJs6bNmzhz6tzJs6fPn0CDCh1KtKjRo0iTKl3KtKnTp1CjSp1KtarVq1izat3KtavXr2DDih1LtqzZs2jTql3Ltq3bt3Djyp1Lt67du3jz6t3Lt6/fv4ADCx5MuLDhw4gTK16M9RXjx5AJRlI7GS2rUmk7OUYraa3mzGlfYUZbavPZykpbARpk8NQf1gIJAZodubZtqiukVIFCkJUCKVcUsIqlgMqVKwxNn1VuljlZ52SHKwU0ReCKTwMHUREoxQ8r3g5R/p9OW2o0VlWZ+GyCPvAzWvFHr2DhDjsWq+GtVnQytAIKFUMMmXeWgGYRSNUmSdRQww10KGQgWQ8SRYUfAl3BRUGGODGfIVecQsgB0t0mIkV8KDjDDDlsMiJLW2QhEBWFEMSGEwAWNCFBknQWSySdxNJJJKbEUookw0UyiUCSTPaKJD0OidmPjkXS2So/QBLLkk1K8uSRseR4JZP2RYJZKZNEWdkkPb4SiSdCEuljJGYKhGabmHkiiWOdcJnkl2NOMlwnd3ap45xOCgnnjjruiSWdPgYayZk9sqJlm0Xq2dmipYjZaJSDZpnlnzo+eiWPQk5S56RSyhmpqZRuuqOS/mAWCmiUelZiX6yRgCpQqphOOiuiu6YpCZuyHmrKHTXIEEMNOfDRySqGcorZJLAGWaxjXsYyJyuaZvpnZYrG6qulwRo6pps/7qokob46WhmphQ5ZZKKTAZplhD0Zwptvw2FHiBMyIhfLdQvpyBETOKiIFHzP9XjVKyXCAEMNO6CSkHvjLSWFEwhcyIoBsVxxgAIksxGLE1AoUEVyHyWRMFIhLpfVJkr44IINdbSSEHti8UzUKTozdErMKrmsyYp4vcIJBDaogvRSbIK0xA1HF+XwcvhOtUAOeTjo81dXvyfS1FUPxfBYrAS5VQNNEKEQxmadjRjZRH39dEkTIFKD/t13h0V333FZcMYOdwBulMEi0XFDJkLJHVZ5XHEgxhxDXMw3V4hnTNIciwflOFivRK0VCF9sUoPFB8Fd1ueJ0WED44azJUIKsfBgR+xsKY4J7mqdUEIsdAgx1+Vdse6R67DzZHxXpYieVQohxIJKDdAapDpZy3vlCfEaub47T2GX9QrRV33xgUA9NGhQaeQ9XYcNl/BuFhkXCFRHD/L/RL5JdtjAx077QxtXzkABgaiiBgojCPe0EsCxZO8j/ftfTh6oldBxJRESGIgQ5mC9BWKFgltpYEkimJOsfeUVHlSKIhowEDvwwCAiBIsJa0PC/H3lEQsYyAETaEOchG8l/newwR5u8sOxvKJ6WhHFAkahQQ4SZBUptIra0gJCkASxazWp4lVYUUSrLCAULXwhQa43lsw1RyZ5sAEWZxLFHkqEAYzQIQIVqJY2KmYPNiicG7PigEQQZIN7vIkWQ4JHPcZkkFURTVcmkAaCuJAg7ENLF8tYEzze7pBpuUxXBEeQHbbHjlNB5GMwgbNAVqUDYiiIENRnSrWQsg6tlAoIvFCQR8YSkzchJStZYka0OS8rI0BBQTzpI1BKRZRVmSRMMnEDJ7JEmSecYVR8Z5AfqC+SZ4Gm4Zjphls2xQvRK0gdxOjNlsRwmTdYAkvO6RVjFkUMHDCIJ90ZFXZ6pZc1/tEEDpSwEmRK5RXajAr9DmLNYlJxLdKEiT75mZKEbuWIXUlDAQ0yTvvQ8ykONdwmcICEch5lDQ44CDE92tCfoIKjKMloVlDYFUUwACHWZMVFnaJSrfgTJRs9wkluChULcgWHCLkfGcWCTyMGZRU5KIJJ7PnQmQZlFEs8yPSQeBamyg8VO3AbSYOygEcghAew3OpJAmoTVexgCE61XmioqhUGKAIhdOhBWpdC1q4UdSeq4AER5jqQu56wrk95wBoQsokZsBV7TFFNfQbCitcwdjVTeUUP5CpWnlCgkQjZAR4qa5DcVOEJvflNcIaTmyuAJyqS7YHTOIuTC4whIUvQ/iprYwEI8BBMIIBYWSym4AfqWAc7CuFpS14hBB6stiPCdYomu4LKhMTBBnxFil+BgoX5xEIK9WHFKaxDierSZyGAzckQjOuR5DbFp1wJAS0R0gmuiU0pVPhDhS5EkAzN50YhM1lViJCDw87WJSU4gUKUoNTZtuhFMRoIF5xACYEcOBYwGuO7ItWtXL1JXeXi1pgOlaodpanCunqVQCpB4Q3jacJ8MtRwgpAiEXepxNG6sIdHteEQi2rGajKxjEmV4zDZGMU9zhSeQiWsGscYUOUK8qHSdeElYUbDsbCTYygBrg8bGcnAigWJw6RjJvMYxI0SiJdhLOQdW1nFYXYx/o9QIIIKDzkWpLSVkk8sZiC7Oc1ffrKFO2ErNZ+5zFjOs7nSPGYuHxnF3GITt6xaE33ZRzix8BfABmII0PJrIeZ9SRFeppFMLwW9W/ECCCyHAwmujikbU8CFTgEykZFMASbbWMcYEl6euIyHFqn1w2rKFDF0ICGlQUJHzaJrnXwiaAsBGleUgIOy/ZclZ6ifQvaAg2d7hNE++ZtFsP2w6PYkDRPY2ZVs8L3orMXTMeFcuSmC7qSAWiuJCCl7HVMEdZ5aLbz2ibotkm+reJsnjHgpQqSThxwUyNpwvUH8EH6SUCwA2Qh5RQ2czfCK9Nsn7zN1RC4eWW4vpRVRPUgk/ofgTLFwnCqHksoVJ9JumP1SK12dt/3IKZaWyw+Pm43Iv4+y85004K0HMc30jhuWnuPOkhUfyQTU0BDbJT3XV9ElRGxulFdMkSsUOANCTGGa2JKl2FeZLlO4+RCqFwWgX3GtzAWSCeg68OkKYSZD4b6RD3zBITgYIt3Zos8k7F0j6nUIEvz+94eI3SkbHbZCDv/Ql2clwGsXCLXfrhawHwWpBU5In6t68qNADyHYHHfyvrL5whsEq2g1/US+MGqHEKGbql+I0XFiViGwZ/ZuHEM8g24QW4Lu3Li3SWqhY3aisMLxWDlDBSIvkOmh7ivFL0rneV5cogtk+k9RpFfA/j1w5zhdhrFXCHHJG/6FrBAilCs/6Hs2hIpBMjTBt4kjcniQ/WnC7V/BPlSi35MitBhJmaR/Q+FwENce+5N3YMF/QyGASJEEVCMQHpdI8VcTIAdGBsEegwcWEUh3StA56mcQC+AIEFFwH1gQlhcVruMsawUWPydy0CFxuKYVJ0gVpUcW79MGoTGDSMFIqRNAPRBWxVOCDbFyQlgBWod+lSOEbIF0JcgBrwURmbA3SqiDUdEJfGADuyQWxwcWpHMQ25MQpeYVCtg4lAFnNzB3PcOAQjE7qcMzGRiESugQzER44UdNEUGCQjiBQ5F45fd5Fyh+c1SCNVgWXCIQm5BU/lqIfFhhPj2oED/YFYUoSZU3TGelhzShfV4xUOuzEHMgPFxBhbGXV7ZnemdgAROhCVL4gWq4FPsjWeTHFSz1FWvwALy3EGH4UOdWhkFHBO4Hi4p4FS7ViApRBGj4QWuxgVSRNf4Xgw8DFkAFQwxxBzSHFavIEq9AGwcBCAMhG4OgjYLBbBTHcFBVgA4xdLFUWqclEJ9wBZNGHMehWw6CbwzhOguXFdX4ExW4iQxBOFtxjykBCFLwWwQRBU4wad9Rdrq4EO+jd1cBOWCxAHFkguekBIpnjErhXdfljZQ2afwBBVJQI+JXRw6BBzaQc80IFn30hwwxeRW0FPhlIRgC/lqxwCGfUAggAkkO0wmYETqYsQqdMByl4DCesJPPYh+fEZSOEZRiRpTQgpRCIpSjoZNX8pNCEjVK6SNBAlBjcpQ5mZNEORxclJReOZWOEZZPKRCeMJYAhQcz8AZWOZZR9mRcKZZLOZVgyZVixiYABS1cBJRCmZNq0wnb5ZN06SNE+ZVVKRCmoJZSaZafIJRRYwpEWZhpWZcA5ZeFKZWGaZRimZlR2ZR46SNRU5QAdVlcdH2AOZWTKSSVYAOb0AmBuZpOeZVSuZec+ZSOYQqjyZSxcAqhqZmN+Zu7OZV8GZqV6SPFaYkY8WARVl/tOBDxpUCbwVKUYBrTeZ0CEYvW/pmd0zkQ2tmdV7IZMpWd3Jmd0qGd3lme4amepnGe2LmeO8Ke7+kYl0AD9vad7Jme8ImeEJif/UmeAIqf+6mf/Hkl2Qk5sRgi2xmg7gmfBrqf4PmgAlqg8ymeEdqg+3kBZGAf3FkaE8qdPWAHBaqg8umf1jmfJQqhACqhKPqf+JlyD0oUlfZo/bKRAuEHAnNbCcF4WwFCzMQEifSLV9FcJsg3biBbVzGGIrExB+AxICMQGSIQrYAyKqMY+lQEyokYs1QRmDADsaRsDHEfjIFVe/VsIyBgFfE65aekP8GjncQDlAUVpyCkVmGHY3Q5JJcVbiqDCSkR42d9n3Z1XQFO/utzOXXgiUkahyLxCu33fGLlaxaBilm6IpOqE8a0aeHobmGhiXT0ELfob7l4UBgxNevmbv7oEwTUQQ9BBPZmFWz6E6dqFNOXcUyBiV4RbzBEPHaAPw2pqCYRRIZUTgF3EZIae8g4FfqHR1lYdWHxjJ36EDegccgaqu+1EZhwhknhk2EBVUx0pxDxeomqFscqFeNaEPqkU0hRqTERc9IJEYe6Rb6aEqinrngBR8SaioUHik+hr500WfQqEcuFkoMFSQukcFbBr06xpxb5EcTVi0KxhWFxWRIZEUMAeyi3Fv+qFfzFjPmjoRjxrvE6GB2YqfIjORjRdqo3iGShsBIx/gfwExQQCxaBR7ASQW5VwbJXgbBMgbMR8T5r1BO26hVnWqQSMQTL+hQ8G7IZoaxuhAIjkBFzgKR0l7Feca10KD+slxF8cAOm96o9EYkloQlJRbXrQ6dWAU9Eq3OBKBVgaxZm2xU6uxCowAOjaKnbtanSNhCh9xA8EKxQEbdKKxGq0AOvaDjclxFJcLVwV65QwbgSsbGWKha42hsUIY1U4bhI26cq0YGlOhMx+xXD6q0SYTpkyxFe2xOxKn0vsZA3EYtg4XAFMZ4T8alQkbqBCxFBdEl3A1WioBF5une2u4AxcYWtKhOu+5BepbcVQQdJGBXBazaa2xJzSBOf+xVu/iW6EnGtoXSMIikThyi1L1G6KyFYz+qna5t9t5sTc6taSJN1G9G3ewe4SSG/E8Go/+cSQesVaqe3bYQExdgU9Lsw0RsTDliPLLEKb1sVdoe9ElEHP3BM6csTrHsbM5sRKBvBkJFGQBgZkLcR51txp7sTSTsSVmuNAfwTKBBOYmZHO/CzTTHC2yuqOfG98Ze/XZG1A8F1FuG/URHCHbEa0KGRrABZ/7W+gIoYuscRIDsYx7GNz+kQpSWTA7GO7bgCUXAFUixuoQG0Q5ADJBsS4qsSyscRF4y+MpEFAyAdT7ACEVFbAjkQUFCQuBUFb5wQPjxBP+GAnRsSAQsWapBB/gZoETC4fzNxCgJAIbEwAPT1EBiJXRgyaY28WCInjz8xwSQxPmORCCzEvxehWbVLE04AWoCQxhHxkosMpZNmypA0Gp8wGqUwHKJRlq7syrDMPhDrkAj6ylcyGsuFy7UsyxA4y9f3y0IizEJCzL1szLocy/ahzK7sGMyczLtcy9enzNJhy+yDy7TMlm8AzdlMy8O8y0lpzeLczNV8zse8y8GMzrqczrdMztonzcVczsslz8uMzex8zcMRutp8zqIBls58JUUApPYMzszcz/P8z+pszgl9z8A8zxB9z+sc0desnH4wAK3wBFnsEMyZYDYaC8wJkgyty+WBzPDMPtrn/svX583jnM/0/M3qaNC8TM5C8tClAFwqDVA2LczMnNInvdP+jNIz7dJby08+Hc49HdDpHM0kLdOyDNRLjc8RvZNCDdAufdA/zdD2TMyN8HDFnJT6vNBbLRBLAATNDNUOPdLgXNPlrNJsfdQU/dItPc8lXc4t0QoDcNEa+RCOdmnANZMcaWmQtngDzBOH+APeVr1ekY9ihox9YHBPccdwbAAFQBGy5qQffV1O0KQB0r1CoQpCsAP+pRHH+5AiuKIVgQozEMYVgbkYMQgAsB0UcWwN8dd0VwQeSBgtyBG5LRidAAAn7BDPGxTBHRKuw5AcAVFjsXScfBEhuq800QpS/sDGMyHZgoQUaXS0F4F2Y2GEn4QRPOwUkv3bCtBgGDwR11qRf+GEHbHEgWHb5z26OyAER6wXC8wRfFBtSVfcQwHDMgHaDnsRiu0VbIiWxjQ9rD0R/t3DCd4SCy4TmzZ6FfFuX1ECvyNmox0RN7DHR/Hg8S0TLovcekGoHfHcH16rT4Es2l0XjNgR4Y3ihV1zUIEJOIClE57AVTEGeTtUEUEH4CvAkwgaULG+jgoRNtwVY6yOGpEHO+AU/H3iKPEKEZ4XahBuHVHGCOfaTKHlNRHi9UsW5/efGCFx9V0UXL4U1m0TaQ4SaVS8DdHHYDF/350RG94Ua14Tww0Ueb4S/plQ49yz5zihREHDuPDLFIAO5SdB5HTB2BxRBBZrbYfOE5GuElOeHGdOFAxw2ntbEUuQeUox6Ttx59WtFS7rwvWH41XRAH5kUBmxq3bOvVusFWlUchFXFjwYoxiR3+eF6FjR52X6FpzUEaZTcU9O3F0xuDvAsUfeFSbL1hkhcUV+FMXuOTFuFVMjrRCI6lSxpayeEWqK5ry+FUFE62gxArTjETuA7eHeQ9f662lhpxxh4tbm4a8eFqjQA8nOWNo+FSTO4xJRBORuFKI+E9P+EyoLi0gQrQdaFpAaCzqsEUqguEdx8OtuFf2z4l+h4x8RtQzX4MNlFpfg5x5/EhKF/toYcQe8qhQjzxIDj0tlsQk8wAOWUBaT6+8RwQeQrRQtHxP7fhU9TxWvcAQKPxbB6OwZob10VfFiQZJu7hXOyhH3x3CXjhSgnhOs0OepBxbciusYUViGTq2a0xw9AtpeHBYxZ/MQoQozUOZkKK6eLTNkbQO62xXXW9oXMQPRbuZKbxZ7cAPqrRUp6RETh3A/XxWF33EFsQk7wAMZThUSu+kW0dvSvhb0zhQ77xLNcxC4re5V4bFUuRHNAu4fh429QcSx0I3deBcrPxOuE/BUQaQd0cJ30wLAkY4Dc8UyqQDvuPcLgfWN/xTcHvumbhsAWce+NTCfcJDhoRaVrxRw/l4Qr0AEtEsVkPfwGiH7StH8K9HIGomRVAAIHfmRDKH9SXH55iSoB2EHM/DoU3ECItDtGbEDIn44LonIV6BfEGb/bECTNhkzCUqgAPEqVqxXAgkaRGiQ4MCCDBMydBhxIcSDEhvGYjVw4kWOCi0+xIjwo0WNIDmW1Jix4kqFJkVS7Ehx4kaTI1mCvJTjxyaVJ2W6/GnTZ8ybMEGqpBnri4eaRkkORPpqxx6iF3sCxYiy6MyWEqMKbapR7FiyZc2eRZt2rJYsA6kU0rilbay3Yqn4EVuq08BOpQj2/StQL1+/sQC/6iQ4cazBhv0iFshqcePDgCVn1KuY8F/Mk/fG/vL0uZOpv489bza8KjDjz6E3I/abmfVAT6Rhc54dy5Ro0501O94s+ffu151OxVp1GrSn4oI9/a6ckZWnVUVuyKHNuzRy5ZVjdycNHHJuyqaPJ//tGrjwwGI4aL+NOGPy7MUxe8qxpzxt08Phd7oPOtN6Y03Av1Qbjz7DbItOLQcfhBBCQ57ASIGMPollwgpZ8eOKgVbAEK1IIiSxRBNPjHBEFFdkscUHX2EOQjtsmMNFG28ci4wLDPOoxfxwBLJEFYMk0kEpnDiAC4wKGOhIBZR8xQkoFKhCrc+KxDLLsa7UsksiXykMwkx0QsVLM8lKgwLGemRxqjO75PJNLE9p/qUsOsU6BSk59+SzTz9xVIUIHPj4M8s1HCDyx0IXZdRMNhtt9FFIF5VUrTpsSGLSGxlhYCYbcbhE0xX1FJXRIUtd9FRU/SwlzBI14WEHTVY98ZEFeMTRhllpTZHXRV31dU9gg31zWBKXsKEOYh8cZQFRWKm0xFdmKHPZtIy1Nlttt+X2LD5wGKLabsVa4BEgUakh2nHXLRJbdm10990WST1RlSFu2EPeBhS5yEVNbJB3oHgDXlESgoNU9WAWp7sRD0zfdSCRxWzE5AaCE1Z44YxvpHfjE9UlUZMdZF2XgjNAjnCPHQhG2WOXX4b5LCVsoGPcC8YA0o4eYubZwTh7/n7wZ6DTApPInHrYhNsPxCilZQeXQIJgoYdOC2Oqz7L6arIY/nLmmrUNIYWJXRzia3mz1jpttbn9FulsSzgByB3yWLtuu++mFIlkrU1hBCB1xTtmgwMfC+21YfTy23CJ/SKEsVncJN2LCSdr6rsNV7toL1Wxzo5gx+DgE6fT+vZgyylHPfU+87hBiKRpPaMCHOkYQvWNR7f9atxbVGXmJWhVYwIchTBb3t2HHjx1zNNGfE9MRsYE1UQceHxFGzI5eHm1B1779MO5L/KVJWpA4ngtHVkAfAczAfjg8HKHP/49NekBB7onDeVWG+kQQn6C1dcaAK/WMTldqgjiWlQr/hYACvONpWwKE+DVkoc67ekuRoVCBRFoBKkFHKKBJbmewioYQNtFcGgE5BMfdqCoQjFgES6yw8og6D8a1vBMdmjdrvwkATW4qAfFs2G3Loi6EKHuFSj0kyqSgKkPsqgCZWhi5FSRMe/VbYKUG+HQmieqTNTPc3y6ABiauITaZSyLVGviy9LosjViyQ444AGh5PSBL7DoFTe4XxD1uEcavoIONhAC9s4UNhbNgQd8XFcV1aZIrWmOV72zARFe16USmKCBqrCBHDNWidxdkXBn7BnXgoUKvSEBgUXyAlNQlIQybsyTiIRlLFGkiSHYQAmnBNIXQDBLwMnSl7+kXCZq/pmESQJpDDsykVSUAExtgbJnr7wbK0wIKU0UoQZDECSOzkABJJ6FCD1o46qgiTdGas2ZMZPmujaRhBoIIXo3UoMEdkcHHOBShMzEZz4fhIqZCeEOa1REA0qUBxu8U5/E6ubaEpq5gKmCDjm4gRKKaSJbkeiPeVRj7s4Zs42+LJ0H4wMRasADO5hPFAs4zoOScINQ8ayjogIEIMoSU420gqZqmWbPchqzI24MFXPAAQ6SoMkIlctBmxBCDib6sp2WagVSqAIUxrKCKUT1Q1K4glQPutVF8cE6N0BCvl7EAEekpXfkmyJXiwQIrTqhiICYwodOAdcPFdEsTYUZXtm4/tB38SEJOLBBEe6wVLGoIhMNEMQmHoUKOtygBzoEml4hhQUsNEmmA6GsWwaR2VhIYRBVs91LPbZFnvFBCTyoAQ6IUIdsMsQOOYjACwIpllXUoQc1+AFGhybaRt1lIFdQklvwEgvgUoENvw3uWfiqteXqrpF8cIMQbFCDHPQACXOoA2oz8ALyzQEJPbjBDHpQB3sCLZyLkotb4DKQ9NKlEO2ti0Y6oaJI7OUVkoiNJDIy34FEQkWS2MsqKJFfgfA3FvUlSCRiE4kC0/e/e2GFghnDYMM8OMEL3q+DBwJgjEi4FBQ2MIdf4WEQ0/czHI7wghvcX/uSOMP9tfCIVVxh/hZfeMKYoe/gEJziCa84FpIYkYw77BdPqMjACBbyhwus4xb7hccfhnGNkxyJF8eiEhZ+cok33OQb0/jAKqpEgCNhCT68wQhEQG0NfFCDGtxgCEqAQx9UEeEMC8S/Nc7ykqXsYi8j+MAQ9rB+DcNkG394v4OTBKIBPeMjB1nCdO6WhlhhoVhgSNIWunQ3y3m1TWtRsgcLVA0w4IIalJRynebVkRCgJFYYoElICq4UWrBqtfB2Y7aG4KcPlgchbEADSIAs3nDNqDuRpdgDOcV51brsLr1CFSogQZ2YPW1q+9ILIah2wMZpt2ET7KOo60IHVNdt04VWdY6kHBiQiTpU/mfb3QdN07vlPW9PEU4NDzi3RpU9udSRlnCH6JTycvc+dp9b1wo7hP4KTm+GUzt/0m64tVKauoMrrLknXEAoKK5Rc1O84lJjACMEbruLD+3jAesX5VjRgERsPOIv5yoF0gDzZbW7Zzbn6cQpt4oLkCF1dkXdtutGbnkpiN0cEMPI8710mvvqFSHwQr+bPnVgiiBuVKcVzjlqcNWV4gQlSJ3Wtz7uc4vdZZ24ttKxvnY9uoftb4c7scqw7rg3iuhnU923T10GNQU9d2Y/e95PLi9PqAFRFKx74nPHKcU3quTmtd2+UV5R1D0eaHd/F+bXJcpTn2IBo0B8CUXfb8nL/otORqXc4Bu/+nExQBGs/5Pq3yX7dfU0dRlxwBpQR3t2aX5dvu8WKwhOuMRU4AyhZ7oRI5/vnisf9s9fWwfqCP09Ad5j1s8YugnXNKgvXHVCXxv408Z54r8Cbsin/p5Kn361oEAE7Id//EmkS/l7icJqJ5y/Awcg0KGfXTYdhB5hBUAYBGkjhJi6LLSonsABPm7Ru/IjAwvwv3V5qiugELGoQK1SgCvgwPqLu+DxQI0ABCmoKxGMq1gAEVbQqhCMlNxJBIFiQc6iggTkLM8yhBWAAilYLxGxHfFrpCEinOdAH9SZhHUxBC3YAroYLuDSCN8iLi4whCv4hEI4AJ0r/gsg3D6u27j8QR0szBZCoKxXgK8dHEO7+AOx6ARFwwj86rI03DATG7IeOzBCmzI9+zI8IzEf2zE2NLRBi7I/i0Ml6zMu68NG27JC0zJAnLIquzNAzLNBJAg+pDLGkAQfE7E89DIgi0QMAw0jg8NFhMRXCLNATEQk4zNDdERMPDJCnMRP0LBUZDQVMQQTaAVWkERGhMMsY0RWxLE/3DFV/MRAu5BSXDQ5dENFfLRJ5JZLOw5Lk6pJ4xAPQUEvFAvL4xlr5KnkSz2COZIkWZJXe5KBkBIEkEYeJLvUMbpTw8aXacAzOTY8gbhKW6h1ZCqSyzt7ZMF81Mewsx3sUxjb/tu9fuykjjMiarQbADlHbSQcenSZeNxHjnnIiJTIiaTIirTIi8TIjNRIigTAshgEQGBIbunIsjAEQ1CbkcSTP/gs5rqpamzJtCFAh2QFlbybCnSCsZACKMgCBUibFYgCCzS2A9ACtXkqq0oJBcAqSqOaDJwqrLpArSnKFayQpHRIqmGrgWiBIjKEm4yFQQA6ngGEKChBscjBodQaukLBIhoEKmiS4QIaPyDBtBTBuASRszzBuhyItbSstbmCyqKLBGQDKqACHdSaGlzJgdCCxJyLqzHMlMiIVlgBfySWxvwtv/QsrenLJjnMU0gpBZBMYimExKSLM3xCjbiCAyAE/kJQACt8mdDcAie8guPiC6liC6oJTS2ATdnUiK30y6HJzSZcQt30zSVMroG4wd4cGjCsrDLELGlkwp5Rztp0L42AAiigghZYAdKETsqSzvgaCDZwApO8GuaMhe7cwaGBL/EcCC5wAkqwG0N4RkrDkEHgSij4zGCBzw2ptErrhE7gQNbkmUzbT0LgSq3JT2jczwNVSqoRUAwphAK1m25kNVcjrhVwgnKkGlVjNSbRCNFMGwnFCFc7TQUgUeHsGRBlBQ7VULVBURE9ABJVABPVmnesqbqh0cC5UbvJ0U+oSqrJ0Y0E0iAV0iEl0iI10iNF0iRV0iVl0iZ10ieF0iiV4dIppdIqtdIrxdIs1dIt5dIu9dIvBdMwFdMxJdMyNdMzRdM0VdM1ZdM2ddOe2QInUAkqqJI3tdM+OYUCoBA/EIDzvNM/NRNAEAAuKAAMBdRD7ZIpAIADQNRG9ZIqAAAIddRJxRFDAIAqEACzpNRNdREDuEkqGID75NQ/lYIBSKkXHdVUVdVVZdVWddVXhdVYldVZpdVatdVbxdVc1dVd5dVe9dVfBdZgFdZhJdZiNdZjRdZkVdZlZdZmddZnhdZoldZppdZqtdZrxdZs1dZt5dZu9dZvBddwFddxJddy5dSAAAAh+QQAZAAAACwAAAAAFwJ6AQAI/gBjCRxIsKDBgwgTKlzIsKHDhxAjSpxIsaLFixgzatzIsaPHjyBDihxJsqTJkyhTqlzJsqXLlzBjypxJs6bNmzhz6tzJs6fPn0CDCh1KtKjRo0iTKl3KtKnTp1CjSp1KtarVq1izat3KtavXr2DDih1LtqzZs2jTql3Ltq3bt3Djyp1Lt67du3jz6t3Lt6/fv4ADCx5MuLDhw4gTK16M9RXjx5AJRlI7GS2rUmk7OUYraa3mzGlfYUZbavPZykpbARpk8NQf1gIJAZodubZtqiukVIFCkJUCKVcUsIqlgMqVKwxNn1VuljlZ52SHKwU0ReCKTwMHUREoxQ8r3g5R/p9OW2o0VlWZ+GyCPvAzWvFHr2DhDjsWq+GtVnQytAIKFUMMmXeWgGYRSNUmSdRQww10KGQgWQ8SRYUfAl3BRUGGODGfIVecQsgB0t0mIkV8KDjDDDlsMiJLW2QhEBWFEMSGEwAWNCFBknQWSySdxNJJJKbEUookw0UyiUCSTPaKJD0OidmPjkXS2So/QBLLkk1K8uSRseR4JZP2RYJZKZNEWdkkPb4SiSdCEuljJGYKhGabmHkiiWOdcJnkl2NOMlwnd3ap45xOCgnnjjruiSWdPgYayZk9sqJlm0Xq2dmipYjZaJSDZpnlnzo+eiWPQk5S56RSyhmpqZRuuqOS/mAWCmiUelZiX6yRgCpQqphOOiuiu6YpCZuyHmrKHTXIEEMNOfDRySqGcorZJLAGWaxjXsYyJyuaZvpnZYrG6qulwRo6pps/7qokob46WhmphQ5ZZKKTAZplhD0Zwptvw2FHiBMyIhfLdQvpyBETOKiIFHzP9XjVKyXCAEMNO6CSkHvjLSWFEwhcyIoBsVxxgAIksxGLE1AoUEVyHyWRMFIhLpfVJkr44IINdbSSEHti8UzUKTozdErMKrmsyYp4vcIJBDaogvRSbIK0xA1HF+XwcvhOtUAOeTjo81dXvyfS1FUPxfBYrAS5VQNNEKEQxmadjRjZRH39dEkTIFKD/t13h0V333FZcMYOdwBulMEi0XFDJkLJHVZ5XHEgxhxDXMw3V4hnTNIciwflOFivRK0VCF9sUoPFB8Fd1ueJ0WED44azJUIKsfBgR+xsKY4J7mqdUEIsdAgx1+Vdse6R67DzZHxXpYieVQohxIJKDdAapDpZy3vlCfEaub47T2GX9QrRV33xgUA9NGhQaeQ9XYcNl/BuFhkXCFRHD/L/RL5JdtjAx077QxtXzkABgaiiBgojCPe0EsCxZO8j/ftfTh6oldBxJRESGIgQ5mC9BWKFgltpYEkimJOsfeUVHlSKIhowEDvwwCAiBIsJa0PC/H3lEQsYyAETaEOchG8l/newwR5u8sOxvKJ6WhHFAkahQQ4SZBUptIra0gJCkASxazWp4lVYUUSrLCAULXwhQa43lsw1RyZ5sAEWZxLFHkqEAYzQIQIVqJY2KmYPNiicG7PigEQQZIN7vIkWQ4JHPcZkkFURTVcmkAaCuJAg7ENLF8tYEzze7pBpuUxXBEeQHbbHjlNB5GMwgbNAVqUDYiiIENRnSrWQsg6tlAoIvFCQR8YSkzchJStZYka0OS8rI0BBQTzpI1BKRZRVmSRMMnEDJ7JEmSecYVR8Z5AfqC+SZ4Gm4Zjphls2xQvRK0gdxOjNlsRwmTdYAkvO6RVjFkUMHDCIJ90ZFXZ6pZc1/tEEDpSwEmRK5RXajAr9DmLNYlJxLdKEiT75mZKEbuWIXUlDAQ0yTvvQ8ykONdwmcICEch5lDQ44CDE92tCfoIKjKMloVlDYFUUwACHWZMVFnaJSrfgTJRs9wkluChULcgWHCLkfGcWCTyMGZRU5KIJJ7PnQmQZlFEs8yPSQeBamyg8VO3AbSYOygEcghAew3OpJAmoTVexgCE61XmioqhUGKAIhdOhBWpdC1q4UdSeq4AER5jqQu56wrk95wBoQsokZsBV7TFFNfQbCitcwdjVTeUUP5CpWnlCgkQjZAR4qa5DcVOEJvflNcIaTmyuAJyqS7YHTOIuTC4whIUvQ/iprYwEI8BBMIIBYWSym4AfqWAc7CuFpS14hBB6stiPCdYomu4LKhMTBBnxFil+BgoX5xEIK9WHFKaxDierSZyGAzckQjOuR5DbFp1wJAS0R0gmuiU0pVPhDhS5EkAzN50YhM1lViJCDw87WJSU4gUKUoNTZtuhFMRoIF5xACYEcOBYwGuO7ItWtXL1JXeXi1pgOlaodpanCunqVQCpB4Q3jacJ8MtRwgpAiEXepxNG6sIdHteEQi2rGajKxjEmV4zDZGMU9zhSeQiWsGscYUOUK8qHSdeElYUbDsbCTYygBrg8bGcnAigWJw6RjJvMYxI0SiJdhLOQdW1nFYXYx/o9QIIIKDzkWpLSVkk8sZiC7Oc1ffrKFO2ErNZ+5zFjOs7nSPGYuHxnF3GITt6xaE33ZRzix8BfABmII0PJrIeZ9SRFeppFMLwW9W/ECCCyHAwmujikbU8CFTgEykZFMASbbWMcYEl6euIyHFqn1w2rKFDF0ICGlQUJHzaJrnXwiaAsBGleUgIOy/ZclZ6ifQvaAg2d7hNE++ZtFsP2w6PYkDRPY2ZVs8L3orMXTMeFcuSmC7qSAWiuJCCl7HVMEdZ5aLbz2ibotkm+reJsnjHgpQqSThxwUyNpwvUH8EH6SUCwA2Qh5RQ2czfCK9Nsn7zN1RC4eWW4vpRVRPUgk/ofgTLFwnCqHksoVJ9JumP1SK12dt/3IKZaWyw+Pm43Iv4+y85004K0HMc30jhuWnuPOkhUfyQTU0BDbJT3XV9ElRGxulFdMkSsUOANCTGGa2JKl2FeZLlO4+RCqFwWgX3GtzAWSCeg68OkKYSZD4b6RD3zBITgYIt3Zos8k7F0j6nUIEvz+94eI3SkbHbZCDv/Ql2clwGsXCLXfrhawHwWpBU5In6t68qNADyHYHHfyvrL5whsEq2g1/US+MGqHEKGbql+I0XFiViGwZ/ZuHEM8g24QW4Lu3Li3SWqhY3aisMLxWDlDBSIvkOmh7ivFL0rneV5cogtk+k9RpFfA/j1w5zhdhrFXCHHJG/6FrBAilCs/6Hs2hIpBMjTBt4kjcniQ/WnC7V/BPlSi35MitBhJmaR/Q+FwENce+5N3YMF/QyGASJEEVCMQHpdI8VcTIAdGBsEegwcWEUh3StA56mcQC+AIEFFwH1gQlhcVruMsawUWPydy0CFxuKYVJ0gVpUcW79MGoTGDSMFIqRNAPRBWxVOCDbFyQlgBWod+lSOEbIF0JcgBrwURmbA3SqiDUdEJfGADuyQWxwcWpHMQ25MQpeYVCtg4lAFnNzB3PcOAQjE7qcMzGRiESugQzER44UdNEUGCQjiBQ5F45fd5Fyh+c1SCNVgWXCIQm5BU/lqIfFhhPj2oED/YFYUoSZU3TGelhzShfV4xUOuzEHMgPFxBhbGXV7ZnemdgAROhCVL4gWq4FPsjWeTHFSz1FWvwALy3EGH4UOdWhkFHBO4Hi4p4FS7ViApRBGj4QWuxgVSRNf4Xgw8DFkAFQwxxBzSHFavIEq9AGwcBCAMhG4OgjYLBbBTHcFBVgA4xdLFUWqclEJ9wBZNGHMehWw6CbwzhOguXFdX4ExW4iQxBOFtxjykBCFLwWwQRBU4wad9Rdrq4EO+jd1cBOWCxAHFkguekBIpnjErhXdfljZQ2afwBBVJQI+JXRw6BBzaQc80IFn30hwwxeRW0FPhlIRgC/lqxwCGfUAggAkkO0wmYETqYsQqdMByl4DCesJPPYh+fEZSOEZRiRpTQgpRCIpSjoZNX8pNCEjVK6SNBAlBjcpQ5mZNEORxclJReOZWOEZZPKRCeMJYAhQcz8AZWOZZR9mRcKZZLOZVgyZVixiYABS1cBJRCmZNq0wnb5ZN06SNE+ZVVKRCmoJZSaZafIJRRYwpEWZhpWZcA5ZeFKZWGaZRimZlR2ZR46SNRU5QAdVlcdH2AOZWTKSSVYAOb0AmBuZpOeZVSuZec+ZSOYQqjyZSxcAqhqZmN+Zu7OZV8GZqV6SPFaYkY8WARVl/tOBDxpUCbwVKUYBrTeZ0CEYvW/pmd0zkQ2tmdV7IZMpWd3Jmd0qGd3lme4amepnGe2LmeO8Ke7+kYl0AD9vad7Jme8ImeEJif/UmeAIqf+6mf/Hkl2Qk5sRgi2xmg7gmfBrqf4PmgAlqg8ymeEdqg+3kBZGAf3FkaE8qdPWAHBaqg8umf1jmfJQqhACqhKPqf+JlyD0oUlfZo/bKRAuEHAnNbCcF4WwFCzMQEifSLV9FcJsg3biBbVzGGIrExB+AxICMQGSIQrYAyKqMY+lQEyokYs1QRmDADsaRsDHEfjIFVe/VsIyBgFfE65aekP8GjncQDlAUVpyCkVmGHY3Q5JJcVbiqDCSkR42d9n3Z1XQFO/utzOXXgiUkahyLxCu33fGLlaxaBilm6IpOqE8a0aeHobmGhiXT0ELfob7l4UBgxNevmbv7oEwTUQQ9BBPZmFWz6E6dqFNOXcUyBiV4RbzBEPHaAPw2pqCYRRIZUTgF3EZIae8g4FfqHR1lYdWHxjJ36EDegccgaqu+1EZhwhknhk2EBVUx0pxDxeomqFscqFeNaEPqkU0hRqTERc9IJEYe6Rb6aEqinrngBR8SaioUHik+hr500WfQqEcuFkoMFSQukcFbBr06xpxb5EcTVi0KxhWFxWRIZEUMAeyi3Fv+qFfzFjPmjoRjxrvE6GB2YqfIjORjRdqo3iGShsBIx/gfwExQQCxaBR7ASQW5VwbJXgbBMgbMR8T5r1BO26hVnWqQSMQTL+hQ8G7IZoaxuhAIjkBFzgKR0l7Feca10KD+slxF8cAOm96o9EYkloQlJRbXrQ6dWAU9Eq3OBKBVgaxZm2xU6uxCowAOjaKnbtanSNhCh9xA8EKxQEbdKKxGq0AOvaDjclxFJcLVwV65QwbgSsbGWKha42hsUIY1U4bhI26cq0YGlOhMx+xXD6q0SYTpkyxFe2xOxKn0vsZA3EYtg4XAFMZ4T8alQkbqBCxFBdEl3A1WioBF5une2u4AxcYWtKhOu+5BepbcVQQdJGBXBazaa2xJzSBOf+xVu/iW6EnGtoXSMIikThyi1L1G6KyFYz+qna5t9t5sTc6taSJN1G9G3ewe4SSG/E8Go/+cSQesVaqe3bYQExdgU9Lsw0RsTDliPLLEKb1sVdoe9ElEHP3BM6csTrHsbM5sRKBvBkJFGQBgZkLcR51txp7sTSTsSVmuNAfwTKBBOYmZHO/CzTTHC2yuqOfG98Ze/XZG1A8F1FuG/URHCHbEa0KGRrABZ/7W+gIoYuscRIDsYx7GNz+kQpSWTA7GO7bgCUXAFUixuoQG0Q5ADJBsS4qsSyscRF4y+MpEFAyAdT7ACEVFbAjkQUFCQuBUFb5wQPjxBP+GAnRsSAQsWapBB/gZoETC4fzNxCgJAIbEwAPT1EBiJXRgyaY28WCInjz8xwSQxPmORCCzEvxehWbVLE04AWoCQxhHxkosMpZNmypA0Gp8wGqUwHKJRlq7syrDMPhDrkAj6ylcyGsuFy7UsyxA4y9f3y0IizEJCzL1szLocy/ahzK7sGMyczLtcy9enzNJhy+yDy7TMlm8AzdlMy8O8y0lpzeLczNV8zse8y8GMzrqczrdMztonzcVczsslz8uMzex8zcMRutp8zqIBls58JUUApPYMzszcz/P8z+pszgl9z8A8zxB9z+sc0desnH4wAK3wBFnsEMyZYDYaC8wJkgyty+WBzPDMPtrn/svX583jnM/0/M3qaNC8TM5C8tClAFwqDVA2LczMnNInvdP+jNIz7dJby08+Hc49HdDpHM0kLdOyDNRLjc8RvZNCDdAufdA/zdD2TMyN8HDFnJT6vNBbLRBLAATNDNUOPdLgXNPlrNJsfdQU/dItPc8lXc4t0QoDcNEa+RCOdmnANZMcaWmQtngDzBOH+APeVr1ekY9ihox9YHBPccdwbAAFQBGy5qQffV1O0KQB0r1CoQpCsAP+pRHH+5AiuKIVgQozEMYVgbkYMQgAsB0UcWwN8dd0VwQeSBgtyBG5LRidAAAn7BDPGxTBHRKuw5AcAVFjsXScfBEhuq800QpS/sDGMyHZgoQUaXS0F4F2Y2GEn4QRPOwUkv3bCtBgGDwR11qRf+GEHbHEgWHb5z26OyAER6wXC8wRfFBtSVfcQwHDMgHaDnsRiu0VbIiWxjQ9rD0R/t3DCd4SCy4TmzZ6FfFuX1ECvyNmox0RN7DHR/Hg8S0TLovcekGoHfHcH16rT4Es2l0XjNgR4Y3ihV1zUIEJOIClE57AVTEGeTtUEUEH4CvAkwgaULG+jgoRNtwVY6yOGpEHO+AU/H3iKPEKEZ4XahBuHVHGCOfaTKHlNRHi9UsW5/efGCFx9V0UXL4U1m0TaQ4SaVS8DdHHYDF/350RG94Ua14Tww0Ueb4S/plQ49yz5zihREHDuPDLFIAO5SdB5HTB2BxRBBZrbYfOE5GuElOeHGdOFAxw2ntbEUuQeUox6Ttx59WtFS7rwvWH41XRAH5kUBmxq3bOvVusFWlUchFXFjwYoxiR3+eF6FjR52X6FpzUEaZTcU9O3F0xuDvAsUfeFSbL1hkhcUV+FMXuOTFuFVMjrRCI6lSxpayeEWqK5ry+FUFE62gxArTjETuA7eHeQ9f662lhpxxh4tbm4a8eFqjQA8nOWNo+FSTO4xJRBORuFKI+E9P+EyoLi0gQrQdaFpAaCzqsEUqguEdx8OtuFf2z4l+h4x8RtQzX4MNlFpfg5x5/EhKF/toYcQe8qhQjzxIDj0tlsQk8wAOWUBaT6+8RwQeQrRQtHxP7fhU9TxWvcAQKPxbB6OwZob10VfFiQZJu7hXOyhH3x3CXjhSgnhOs0OepBxbciusYUViGTq2a0xw9AtpeHBYxZ/MQoQozUOZkKK6eLTNkbQO62xXXW9oXMQPRbuZKbxZ7cAPqrRUp6RETh3A/XxWF33EFsQk7wAMZThUSu+kW0dvSvhb0zhQ77xLNcxC4re5V4bFUuRHNAu4fh429QcSx0I3deBcrPxOuE/BUQaQd0cJ30wLAkY4Dc8UyqQDvuPcLgfWN/xTcHvumbhsAWce+NTCfcJDhoRaVrxRw/l4Qr0AEtEsVkPfwGiH7StH8K9HIGomRVAAIHfmRDKH9SXH55iSoB2EHM/DoU3ECItDtGbEDIn44LonIV6BfEGb/bECTNhkzCUqgAPEqVqxXAgkaRGiQ4MCCDBMydBhxIcSDEhvGYjVw4kWOCi0+xIjwo0WNIDmW1Jix4kqFJkVS7Ehx4kaTI1mCvJTjxyaVJ2W6/GnTZ8ybMEGqpBnri4eaRkkORPpqxx6iF3sCxYiy6MyWEqMKbapR7FiyZc2eRZt2rJYsA6kU0rilbay3Yqn4EVuq08BOpQj2/StQL1+/sQC/6iQ4cazBhv0iFshqcePDgCVn1KuY8F/Mk/fG/vL0uZOpv489bza8KjDjz6E3I/abmfVAT6Rhc54dy5Ro0501O94s+ffu151OxVp1GrSn4oI9/a6ckZWnVUVuyKHNuzRy5ZVjdycNHHJuyqaPJ//tGrjwwGI4aL+NOGPy7MUxe8qxpzxt08Phd7oPOtN6Y03Av1Qbjz7DbItOLQcfhBBCQ57ASIGMPollwgpZ8eOKgVbAEK1IIiSxRBNPjHBEFFdkscUHX2EOQjtsmMNFG28ci4wLDPOoxfxwBLJEFYMk0kEpnDiAC4wKGOhIBZR8xQkoFKhCrc+KxDLLsa7UsksiXykMwkx0QsVLM8lKgwLGemRxqjO75PJNLE9p/qUsOsU6BSk59+SzTz9xVIUIHPj4M8s1HCDyx0IXZdRMNhtt9FFIF5VUrTpsSGLSGxlhYCYbcbhE0xX1FJXRIUtd9FRU/SwlzBI14WEHTVY98ZEFeMTRhllpTZHXRV31dU9gg31zWBKXsKEOYh8cZQFRWKm0xFdmKHPZtIy1Nlttt+X2LD5wGKLabsVa4BEgUakh2nHXLRJbdm10990WST1RlSFu2EPeBhS5yEVNbJB3oHgDXlESgoNU9WAWp7sRD0zfdSCRxWzE5AaCE1Z44YxvpHfjE9UlUZMdZF2XgjNAjnCPHQhG2WOXX4b5LCVsoGPcC8YA0o4eYubZwTh7/n7wZ6DTApPInHrYhNsPxCilZQeXQIJgoYdOC2Oqz7L6arIY/nLmmrUNIYWJXRzia3mz1jpttbn9FulsSzgByB3yWLtuu++mFIlkrU1hBCB1xTtmgwMfC+21YfTy23CJ/SKEsVncJN2LCSdr6rsNV7toL1Wxzo5gx+DgE6fT+vZgyylHPfU+87hBiKRpPaMCHOkYQvWNR7f9atxbVGXmJWhVYwIchTBb3t2HHjx1zNNGfE9MRsYE1UQceHxFGzI5eHm1B1779MO5L/KVJWpA4ngtHVkAfAczAfjg8HKHP/49NekBB7onDeVWG+kQQn6C1dcaAK/WMTldqgjiWlQr/hYACvONpWwKE+DVkoc67ekuRoVCBRFoBKkFHKKBJbmewioYQNtFcGgE5BMfdqCoQjFgES6yw8og6D8a1vBMdmjdrvwkATW4qAfFs2G3Loi6EKHuFSj0kyqSgKkPsqgCZWhi5FSRMe/VbYKUG+HQmieqTNTPc3y6ABiauITaZSyLVGviy9LosjViyQ444AGh5PSBL7DoFTe4XxD1uEcavoIONhAC9s4UNhbNgQd8XFcV1aZIrWmOV72zARFe16USmKCBqrCBHDNWidxdkXBn7BnXgoUKvSEBgUXyAlNQlIQybsyTiIRlLFGkiSHYQAmnBNIXQDBLwMnSl7+kXCZq/pmESQJpDDsykVSUAExtgbJnr7wbK0wIKU0UoQZDECSOzkABJJ6FCD1o46qgiTdGas2ZMZPmujaRhBoIIXo3UoMEdkcHHOBShMzEZz4fhIqZCeEOa1REA0qUBxu8U5/E6ubaEpq5gKmCDjm4gRKKaSJbkeiPeVRj7s4Zs42+LJ0H4wMRasADO5hPFAs4zoOScINQ8ayjogIEIMoSU420gqZqmWbPchqzI24MFXPAAQ6SoMkIlctBmxBCDib6sp2WagVSqAIUxrKCKUT1Q1K4glQPutVF8cE6N0BCvl7EAEekpXfkmyJXiwQIrTqhiICYwodOAdcPFdEsTYUZXtm4/tB38SEJOLBBEe6wVLGoIhMNEMQmHoUKOtygBzoEml4hhQUsNEmmA6GsWwaR2VhIYRBVs91LPbZFnvFBCTyoAQ6IUIdsMsQOOYjACwIpllXUoQc1+AFGhybaRt1lIFdQklvwEgvgUoENvw3uWfiqteXqrpF8cIMQbFCDHPQACXOoA2oz8ALyzQEJPbjBDHpQB3sCLZyLkotb4DKQ9NKlEO2ti0Y6oaJI7OUVkoiNJDIy34FEQkWS2MsqKJFfgfA3FvUlSCRiE4kC0/e/e2GFghnDYMM8OMEL3q+DBwJgjEi4FBQ2MIdf4WEQ0/czHI7wghvcX/uSOMP9tfCIVVxh/hZfeMKYoe/gEJziCa84FpIYkYw77BdPqMjACBbyhwus4xb7hccfhnGNkxyJF8eiEhZ+cok33OQb0/jAKqpEgCNhCT68wQhEQG0NfFCDGtxgCEqAQx9UEeEMC8S/Nc7ykqXsYi8j+MAQ9rB+DcNkG394v4OTBKIBPeMjB1nCdO6WhlhhoVhgSNIWunQ3y3m1TWtRsgcLVA0w4IIalJRynebVkRCgJFYYoElICq4UWrBqtfB2Y7aG4KcPlgchbEADSIAs3nDNqDuRpdgDOcV51brsLr1CFSogQZ2YPW1q+9ILIah2wMZpt2ET7KOo60IHVNdt04VWdY6kHBiQiTpU/mfb3QdN07vlPW9PEU4NDzi3RpU9udSRlnCH6JTycvc+dp9b1wo7hP4KTm+GUzt/0m64tVKauoMrrLknXEAoKK5Rc1O84lJjACMEbruLD+3jAesX5VjRgERsPOIv5yoF0gDzZbW7Zzbn6cQpt4oLkCF1dkXdtutGbnkpiN0cEMPI8710mvvqFSHwQr+bPnVgiiBuVKcVzjlqcNWV4gQlSJ3Wtz7uc4vdZZ24ttKxvnY9uoftb4c7scqw7rg3iuhnU923T10GNQU9d2Y/e95PLi9PqAFRFKx74nPHKcU3quTmtd2+UV5R1D0eaHd/F+bXJcpTn2IBo0B8CUXfb8nL/otORqXc4Bu/+nExQBGs/5Pq3yX7dfU0dRlxwBpQR3t2aX5dvu8WKwhOuMRU4AyhZ7oRI5/vnisf9s9fWwfqCP09Ad5j1s8YugnXNKgvXHVCXxv408Z54r8Cbsin/p5Kn361oEAE7Id//EmkS/l7icJqJ5y/Awcg0KGfXTYdhB5hBUAYBGkjhJi6LLSonsABPm7Ru/IjAwvwv3V5qiugELGoQK1SgCvgwPqLu+DxQI0ABCmoKxGMq1gAEVbQqhCMlNxJBIFiQc6iggTkLM8yhBWAAilYLxGxHfFrpCEinOdAH9SZhHUxBC3YAroYLuDSCN8iLi4whCv4hEI4AJ0r/gsg3D6u27j8QR0szBZCoKxXgK8dHEO7+AOx6ARFwwj86rI03DATG7IeOzBCmzI9+zI8IzEf2zE2NLRBi7I/i0Ml6zMu68NG27JC0zJAnLIquzNAzLNBJAg+pDLGkAQfE7E89DIgi0QMAw0jg8NFhMRXCLNATEQk4zNDdERMPDJCnMRP0LBUZDQVMQQTaAVWkERGhMMsY0RWxLE/3DFV/MRAu5BSXDQ5dENFfLRJ5JZLOw5Lk6pJ4xAPQUEvFAvL4xlr5KnkSz2COZIkWZJXe5KBkBIEkEYeJLvUMbpTw8aXacAzOTY8gbhKW6h1ZCqSyzt7ZMF81Mewsx3sUxjb/tu9fuykjjMiarQbADlHbSQcenSZeNxHjnnIiJTIiaTIirTIi8TIjNRIigTAshgEQGBIbunIsjAEQ1CbkcSTP/gs5rqpamzJtCFAh2QFlbybCnSCsZACKMgCBUibFYgCCzS2A9ACtXkqq0oJBcAqSqOaDJwqrLpArSnKFayQpHRIqmGrgWiBIjKEm4yFQQA6ngGEKChBscjBodQaukLBIhoEKmiS4QIaPyDBtBTBuASRszzBuhyItbSstbmCyqKLBGQDKqACHdSaGlzJgdCCxJyLqzHMlMiIVlgBfySWxvwtv/QsrenLJjnMU0gpBZBMYimExKSLM3xCjbiCAyAE/kJQACt8mdDcAie8guPiC6liC6oJTS2ATdnUiK30y6HJzSZcQt30zSVMroG4wd4cGjCsrDLELGlkwp5Rztp0L42AAiigghZYAdKETsqSzvgaCDZwApO8GuaMhe7cwaGBL/EcCC5wAkqwG0N4RkrDkEHgSij4zGCBzw2ptErrhE7gQNbkmUzbT0LgSq3JT2jczwNVSqoRUAwphAK1m25kNVcjrhVwgnKkGlVjNSbRCNFMGwnFCFc7TQUgUeHsGRBlBQ7VULVBURE9ABJVABPVmnesqbqh0cC5UbvJ0U+oSqrJ0Y0E0iAV0iEl0iI10iNF0iRV0iVl0iZ10ieF0iiV4dIppdIqtdIrxdIs1dIt5dIu9dIvBdMwFdMxJdMyNdMzRdM0VdM1ZdM2ddOe2QInUAkqqJI3tdM+OYUCoBA/EIDzvNM/NRNAEAAuKAAMBdRD7ZIpAIADQNRG9ZIqAAAIddRJxRFDAIAqEACzpNRNdREDuEkqGID75NQ/lYIBSKkXHdVUVdVVZdVWddVXhdVYldVZpdVatdVbxdVc1dVd5dVe9dVfBdZgFdZhJdZiNdZjRdZkVdZlZdZmddZnhdZoldZppdZqtdZrxdZs1dZt5dZu9dZvBddwFddxJddy5dSAAAAh+QQAZAAAACwAAAAAFwJ6AYdWSlVYTFdZTVhdU11lWmRlWmRsYGt2a3V5bnh1aXN9c33/pwf/qQrnqR3/rBP/rxr/sB/Sqi7eqSH6rib/siT/tCv/tzL/uDb/ujyorVfetkj/vUP/v0n7v1P/wE3/w1P/xVv9xmL/yWX/y2z/znT6ynT/0Xzlx3OGfYWIfoeJf4hws49ttpdJuLdwu6UdveILvvUBv/4FwP8Lwv8TxP8bxv8hx/8kyP8syv8zzP88zv9YzutBzv5Iz/5YzPdF0P9L0v9U1P9b1v9pzfRg1/9l2f9s2v903P993v9w1/yMg4uQiI+TipKYj5edlZyYkJecvYyjnKOnoKarpKuuqK6zrbO3sba7tru+ub7Aur/oxo//1IT/1oz/2ZP/3Jvy0prK0aTS2LH/3qH/4Kb/4av/5bT/57r/6L7CvcCd0+6B3v6N2/iJ1PSo3PO63vGF4P+K4v+T5P+b5v+h5/+k6P+s6v+q4/m07P+87v+y5PbHwsbIw8fKxsrOys7RzdHV0tXa1tnd2t3h1sr/6sP/7cz16Mfo4dz/79L059L/8db/8tvE3e3a2ubd3ejV1ePh3+Hd4+7N5vPB7v7F8P/L8v/U9P/c9v/k4+Tl5e7s6uz27uP/9eP/9+j/+e3s7PLw7/Dh9v7q9Prl+P/s+v/y8vb49/f/+/T29vn0/P/////1+PEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI/gBTCRxIsKDBgwgTKlzIsKHDhxAjSpxIsaLFixgzatzIsaPHjyBDihxJsqTJkyhTqlzJsqXLlzBjypxJs6bNmzhz6tzJs6fPn0CDCh1KtKjRo0iTKl3KtKnTp1CjSp1KtarVq1izat3KtavXr2DDih1LtqzZs2jTql3Ltq3bt3Djyp1Lt67du3jz6t3Lt6/fv4ADCx5MuLDhw4gTK17MuLHjrozURkZLylPaS6TSNlp76dTltJXTesqMdrLSUnz8GPy0R7XAP3xiP55NmyoKJ1KYECSlwAkVBZkVRKFChSFpymqPm1W+fCmfJwJRZBroJ4pAJ3pI6XZo+mz3sqHR/mJO+90olSrXXacilbkUikuBUDCJAoihZdGgmZO9j5Y/0ih6CEQFGgUFogR6gVDxyR8I6FfbgxAOhYUVAtFHEBpKBGIQgAQ1slkqjFySCiaNiOhJI5Zd0khmjURGSompnJgiI5kxslkoPCyyHowypnKJIwJ5uKOIL1p2Yo2TOUIkI5jEuKKPjHjGSJImouhkjUCm0uKORjqSmYo1fqikk0bSCOKHW5LCJJlQIinQmEVeCWKWafLISIpP2vhmlVV++eGU64XIJomW6ZkKnI4Y+aSKUrrIo5VgaikmJFzG6KWPf252yqN4himQoC82eWSbkqYSig+UxklimJY5kqqV/j1eYqaQhy5Z5peTpSlorHT+aeKdcsr6qaNVdgriZLvCmieakalYbFGBLLEecKlM94cSFxaXinQLlYdREjVYkpS3YpEi4lY21LHQeKUx5YQSCGSxngGpUHGAAvgSqAQTCkhh3EdGhIuUg2MRTBUSQixkcFgLD/VJKQ190vBIAVcSoV6UzODZxUr51xESNFhc1LlnsdeVDXMo5HFZJLcbEsgiD0UuwytfhUQQCrHr3VobvxwyURNzXFIlMvQstFowH00XDikrXdTMHMFBAyVCQf2VJ0129QYQCels1oeflfTG1EFZ7VV4XFkiwygIYRI0V2YfJvUkTsOVQxx1szV3/t5twfHDXG9DZjRJcMxAdU9xc4X1V6DIAMpBXpeVeFdZoxTHDHTz1DJ4gUelA94G1TyW6I9dLgnfasWhA+o/dc6RHDPgsZPrWdHu1CgyiFuQ7VfxftXkHsEue07A1046Vj+8YVDkZBWvle8ZCZ/T8V+dAv1ScuRg0PVTUT+b9KyThbvu4ee0+UpzzHDHTecXHIpYyRfkyeBkVe7yS+mrW5Pzvbe/VfYFYd5Y+IcV+rGkDjPQ30y4V76JjO8tDDTMHWbQtAZ6JX5u4QRnaDLBCsKEgFVBG1gAOJDRoEUME1gL2GQyQTnIBIRUEeFXHigQAYaFAloo3yRmADoLbuUH/nBgCxcoYMEd9tCHWCFhWkrxADKwBYYi2WEQXbJC8NgPLDR0m1m2YAEtbRAnlKCB8lriP5qVhQdTNCFZOuGAMqTiEUgMoxqQiJU4aA8tJsCAWyIYkjAigSV8fEogk/LAQSKFEww4g0AM2RQoDq0GR1iJI6Miw7Cg0UeMLMoIODCQSUrFezGpBCRVAkqumKwsdowRWTixAEKUkI4DsUQNigDLqeDucWQBAQjgUsqYgMIGtDxJL7cyzKugMZNDScQCEkGQYmrFkyaRZTBLAk2nVDIsqrOhVj4ggoJUsTlA+WXCSoLMpJSzKI0j31cIwQAN7qaWBRkFDoJgQHge5XNi/tnACORSRp3IEwj13Eg/v0KK951lDTwI6FUGwYBOLE+FRBlFDugpkm8WbKBdsUQMcOkVDJjgINWECWrUMxBStKakqZnKKXSgA4XaMyhM+0oZHODQqdxGCtIqaW9+k5nbUGE7UVmpDtj2UqS8AWdescAWqMKH7XBLIHzwVyqeoIfnRGc6CgmpSk7xgxwQtSNaXQoprkiWPGisK2N4AMQOYlGiVAE9qXCCekjxieg84q3pWVdSgOBVj4RVKdcUyyVssD6uVKALCfnrSqKwBwHJiyAGQg+H6kUgqpwiCDYwaFGJcoRxaqULRKzKhCpUn4FkQQlwTMVoU2GhgQgLRLaK/pGZXgso2AbKSOMxFKiA5YnZ/mkykIhtb3H1KSJZSU2Z8YENxFVbGKnpVlAq7m1l66dhSfe51I2ubbGLXExZd7u89e11YWWm4XoXvNClrXEtg13zvta54a3umQQS3Ol6Yjzqta94tbvb9CJLuPutbX+zq6JUSGIGqQJWd1+b3/YGOLYLptSxrhvf8w64t3iq4X/XA4lMPOALi9qwmprUXaIEQje8yYy1sDWQaE2rYYpdiRAEppEYH8UTzrwKZmgwPKxwoQLdcpcSFCCvT9DLXvhSAIHeFS+GYDQoAVOnRZ7clRxbxTJFMEJWSuEAJ+asKZlY60IexpUj1CBmm/3J/h1qkBUTXAAv52RJ0i4S56KQwqVnS8UpMHeVTiTyX5KBytgyVxEbGyWwYdGiEP5oFRLokSGGvnFUBm0RK1ulzkDzTB1sYBVEDqIhlq6l1E6X5p6cQgZojooIPKCXUPfkcj2OiKulcsqSDQQIY5RKIhhwCIfMeipRmkr+JhJpoixOPKRJ5VR0+ZBiO22CdJAIpkstkMZ99SmHWOZDpv3sGbiQ2jnJwbeh4oF97oXKSZEiRND9vF9DRY2puFlUCLEAdzqE3VtpK1Tk2GzQkHUsOssYnomCARJExNl8C2MkwW2TGhS2KWdoKMMfqeWJzyTLT7nAR/uib6nIsuJZ9Xda/rQ4kDU7hQwOEHO/1YLvpYhTr5Rx91PgLRDDNaUCXJhIyy1ebRwAlOcvCcIclyIGtf5l4FGR5w/qye2X0k+JSaGAFyjS9KpVnSZCNSDCgzLWyzCncRw9CmgLvRaZF4WrfZVffkBDP3EnpRQQGENFzE5ttF8b6CfZWlK2AOTA0P3sQMBB2P8u1qv7xEGVOKtR2OjGua9l6zsRwnIH0nGCEl4pAnT4UfJ4EcgH5fJFMcLP1pMc0CAE40Xx86ctYngkHoFseDfJpo0ygg0QZudUKRwecB9CzZpFdKeW8k9Y6cqL8N4qEjbL5dIAmuNPxYb4HIoIPpARz8Nz2LEfid6F/qJMZma/Ji38fh+LJpQPhMAwzn8+HmYwRbIgGiwkP0gNYt2TXXsfI9b/Sf4Rl4pJ0GDhBQN6SNEZCYF6P1FuG7F/9hRGICd+GzF7P8FQNVUYrRcU0uSAEbEwwfcTHoUYCrgTWSIQlmADnkVQ/yZYCxN9PFEGErcRIYgWJyhY8eRzSNc9+KEQb/A3PXEBS8UR6YeBByFRSweEF0E0NdgSKKdyhSGAA3MQK5V2plSBs8MQ88cTOOcRUsgTH6gT5HJZgtcV7/cV2iQQQgCAOOEFEKCE1bcWWagTxyN5wtc7prcQc3BHOSF1H8GEKYEafIAQffgaqfGHgWFmqUaEDgEK/jFwdzQxRG3hU0AlEJlABSyWCgpAHFKlMmrhPYVDalgRhl4BSjF1E3DnZR6hhyfBB05wVQTRBErAYtqxcvejEJfzcJdmikQxhgJxBNNUE1vwZiCxhRqBV3EliAJhIMUoH05QWgrRhjnxNnQwA9FmiKTHECZnE4wXEsyYEYzlWAWSUwmSCYDQIM10H5fAXlpkLplxXwKBCeT4PugYI+eCCedSjusxHveVjvFIjuxlQuoYI1lDj6SQCbiFjzWkj+vBjvAoEP2IY/XojvaYjwW5Hp5ABzGwBlnTjwjpI+z1kAo5j/tIkAmpkQ1ZjyApj+u4j+YIkvQoku94bP1Ij6cw/h4tGY//mBkz2ZEK+ZHwWJIGeZMj4pESaZMc6SP/uJFCiZMiiUkHmY6QMAOWgAlFSZJS6QlYtZKX4JAqGZXm8j6fcC4YWQJ6dJVTOY9aKZNDaZIjIpTZKBFXQCGspYzFOIkDsY0lRRp3ppSLVFKLZJcbcxx+mZeAOY16tpeL1Jd6uR6kYT2B+Zd6ZpeHKZh2mZgbYxp9GZl5KQkwwGjTeEqMOY2KuR6GuZiiGZgb85mdeRyVOZqVwV6OOZqbmZiJeZic2ZqbORCm+ZinWZij2TOoKRCmGRqcGZrr8TmnIJyMGZy6WZuPGZq/KRCasACKdJeC2ZfU6Zu46Zs1wptG/uFiKVYtLcZieqAtT5UQlfcVxRNGSRBCMQgWuCgQaoBUM7FJIwGMGsFk8kIK9BKXAlEK+9IviiFKQnCEDrhDNKFMvQYXZGYca4kVoDBPAip+NicTzPYY9HkT5YkQEtVSlLSeXxF/C4FrMtF9JHGhXvGDUQFCdgcVnlhlbxMHOggT3ERN0rgSpxB4YTejA2GEMEFv9uYYC1oT3DNjhWhOaXGEVfgSG2Bw5PR4gYYRIENogGWLQ9GeAxEEmskSEdejIlGhNyGlQ+FusMYUtfZ7ECEHq+MSGNCDJeGlOGoQ6eNBM6qjLDFTEzgbP0oTZjdB7deEyAERPNYSVyhMTEoe/h3hf2ZYFAyJbBAhdCxRdGq4pWxYeh0BoETapw/hoixBAYiFEncKhA3KNXFKfimBhhdjolBhqvHEUorYOmwKFDTXEDQApSXBRGKgEqj6FCQKNyHBVV84FF2nqBABBEN3Ej8mSZHaGDMWh9nnNyjBRI3XpmBhZoeDgYmHEm4mNLfaFLlqESDDiYfXqj/hoQ7BZyXhZ4q0Etu6FQSIFulaEZejQD2xolyxc0CwpyJBArbHEu0KrSahpxh4VCXhafxqFv7XgLGHBzRQEtOnNPt6FS84NCT4oKTEoSVKOxsoEspUfCzxsGaRrUzhsQsBCjkwhDsBrj5hZTkApx4xoWQ0/rD+pANQCHRGYLAdcQgMcH9C06ky0XqYpaw7y3YTUYcigYAvobMy0bBWsX+vJ6sLZLI9QaUGoTYS2xDspKXGmolY6xKziBNOyxO+c6QekaQx0bUuCxHpM24TB6IeEXF1ejRkW7IxsX5XGhNjyjkUAQeg2hEaJxNvuxNcuj8ywYB4SrFdAbUG4X8ekYQvdKyWChMjCJ91lzsdEah0W7ZBIbJDxXApyxFikIaoA7KYVxM1OnlF27c4Ia4QUQSHehF4OBOgqxR/SxOxK3reuhLm4nXuygMbMXaya7lEsbVpljEaMYq+qxcIdEQvJbkYwXflE7szgbQlUbCAZLrsAz04/gCvFHGNNQG9VeG8i6sTj2u71GsTrxoRqosRnGcT3ksRqWFAgkgKKTVxIhuzsISpFmGua0EcA3EtEuFTOTUQkTiJKNAEVPC/CWG0MIHAvgkENjCkIqHARSG8FlF7OJGNVjAApLEEKBARTaWKA8EErQhVTeDBidWkPSF6TIuN41sThuuEyjsRxIcT9PkJAhAgqTAAj/UQwihXBcJiO0xSB7HCNSHEsqg+JlG37ocROYC9EBEC1Me1NKEE0sIHGBwRk0UFOayfrGXDWDyOkHgfJlQZNgnGYEwaYXyRZRyUErlIZKyQYzzGbJyTbqzGiVrHb7zGMdLGMZIZpxDGemzH/ngcHgwpxnT8x2Z8x3VcxhS5BnCcx3O8x2osyGkMyZIsx5R8H5UMyZSMxpbsx3HsyI4cHqJsyHicyI/syaDMkH18x5k8yD6Qnq08ySaECAvQa4BMyKPcyaz8x5jcyK3Mx0iMEnowAKWwBAbsEKvVWt8pEMmsISWVxjjmy6SMy6R8yek4zZjMj9j8yYB8yQqJVbcsk9i8y7ocyONszuWMsFr2y5GczeVMzdc8x/BcymlMze/sx418y/icyoeMzvzczjkZz5rszv8cbz3gz/qcGR3QAeVszalczwSd0IUsx9Gsxi1RCgMwzMToECf2Yt6pxdxJLQrBvcA2FJaAAyQL/hK/ehYtfBB5wGkSkW1W27s0wQQGUAAUYZ/zsszXoQQHkMUIQcQLFFE/0KsqPYcWgYiruhAcYG7NWBN+AADWQRFh1hBYhYEzNq3lA3sPwVAzzXIA8LqyxnJHUTi0uBHBLBblKxE6gLYNsQEbpxNiHRGl4AQbPBPr+0FIgUD2mhG32x/cc74PwYJfTdMxcQkAoACpVbwa4X+7yDf26xA8KBdXzdgbcdI/sNRCgwds5hBjkHK1NNdGQdIzMQpFfaMVsdJmgboS0ThIp6k+QdpSYSbs2hRZ7ddCHROsLRGx2hCkGtuWbRVvYMR509YMAXdTF9x29hTp09ccI9gKwbuH/jeoscgUk1ADAZrauQ0TLY0QcAC5B8FEpMh/ZB02T4G5qO0Q8kpMGlEHOLAQ1woUoq3cM3EKty00EowQpfBnLwXBLeHfFjHcZ63eSH0Rp6bZjiYUAM4Sef0SDZ4RCDS3xrHdL9HdCNHbByGwZVN2WUsVlIDdR0jhLsFHm3sQI8BqQiHi9A0S6B0hQjCsBLFrGvtSKt4SNb4R960wC54S0umkJTgQLPt51L0zWTHcTBw68z2lEWSmBmF/RPHgIy6pWYFAuQY4GyEJnU0QMQo0Kw4WH05Rj6E2BcGj1JbkU9oVo6ADOKCs660Vaz0RpxZ2YjsyQ24WUO4z9LdIhDuv/nwUoakggUZx511OEulT5YqBAz3WgYMOT9ed3Yth3KmguOAm21Ah6CIxv+TT5llh4QghBGNkATl3FJZuq2qRfGEhBH+Kx2fx5hNxBFrmqEhh6osuFrBj6IQBsLA963xKFv7n6IRxCtnzBQ8AWHUuOWdhCTmQA5GAuxhxCnWgAzLQAGAAu1/UH2hh36n+e1NbEMguAzAQAT+Q3kJh5roeFM8o4XyBB94OAy1Argx34ysB7ypBCh/+c0mcEZQgAysgAzLg50Yh7yox6igh8EcsIqPAwA7M5xkBCjsgAfweBJoNFAR/EjvO41KOBN5WMBpxCiegATHwBinsq+XeFndA/gNFsO1ngUiF4O8MR+5CThnMcdI5IO5YweoQMQLUx9UDCFGEShlkdQpFkO0KfxEyngo2kOdPwxR8uD3xmwp+EIgMgfIxNBeX49xxMaHXKzQq4BuPuC0EnFOVSByPOvIK8eE/4HtxkW3el/UXg4okbFXbkgmveG8mvBzrOQpBALY1fz0c0E0CgQMDXhSUDhM7LIjCGAV8EB9M4ATOHHI9XzLeIwcxAONWgQkoX7UDwfZJMfgvccWVtcUCggbfGI7K4ZfGKZut+ZmvSZiQifrJaZynr5qWuTusn5uAyZnXiZiyeZjN2frWqfvAL53HOfuxiZ3Jafu4bxCSYAM/YAmw/rn7jcn6t/n7q+/7g3n7sd/7vRn9wU+bkumboVlwtgn4pzT9nsmXrvn8wW+brm/9ww/9RNGWpDUQyQyXk4WUAAmQWpQJQMmSWrSQAHEpVSpPnlKRKnjwEimCAlNdMvjQIKmFChl6wjTwUsZTFTFedAhR40SRnjJdzDQQ00RMDEm1JBhRpESFBj15lDkxIUWGoRxi1JiSZ8NTMQcmfBjKYsOjGVOt1OjSY8icS28WRSqSIslQQmi0UekQ6lCfR6keTHgT5Ei2Qz8yfUpS6tq4bSuWhQv14dxUeDFRPcMgEEgcd24GtdtT7Fm3nujq1TpVpVOtMF/SBbp16UDOnT1//gYdWvRo0pwDLTmogGHK06lJ6aEyEEXK0ZBK38adW/du3qkY9QYeXPhwhLrlzHgzXPly3BWBX9jiuTBz6sJ/V8fO24kSBFkOFhi4XYH3U0qYKJBS2mF29u1Jr3cfX35x3ZRs8AAlX//tiL3JOCjFMxvu2K9A+AqU75MAP1Ows08YQjBCCSekMMJRgqgBjwo31K0CLj6bjkMRR5QQQhJP9MxEFFdUUbc4ZjBixexazM0LCBbsrAZJZBSORh5FvO5HEoMUkkOkeqskBxwqKXI453aDwAvQZmCySd2ItLLC/rKkcEsuJfSSNyRmiOPL3XwsjQsKQDslhvzMJC1MOOek/rNOO0fDowYg3rxzuVIgIAM0UGQoqk9DwTw0PjkTZQ7N3UYBggYCGSWIty0qCK0SGSitlNPsGvEUOyxD7XHR3uiAkdInbyvFgTJCm6QGTkcltcdaG711OUeRxGHJRHcFzQQMRKsDB06BzTVZZZcF7YgZ4GCWs04YOEM0OXSINlvRDtRWvW5zow87SWzQwZI+HcttBA5GQ6IITrn9djRa4wVtXno7Cxe7U5yF1s5VR+NkAUJGCyI5Su29N2GFv8yz3GRB+IA0HOpYuGKLL47wlCLIvDWRBRIhjUqMkwV1ZM4QThgv+fLcE06YSvtABNIsIXRWkzuD12KU7803vlG8/pLDTHRJO4QBkPGU9d2bl2ZaxDpo+MFcRjcYoTQ4gGg6VGSzprdQBEdxFolEB2GAk9J+6JfSre8t+ead6f3E1Pgm6XWSJv/9DAMTbpuBEk/fvlduhXNemJS1szsFCRmK8FpGwctwoJPSKJkhVMG5xjxz6irRoQaK6bQgOqt/0JzRywNn+nD3XhSCzxN3HeMBHEcDIm1KT2ebacC/fYlEUIJAbkW8Oaugi9tO6TvU3eNVvVvcmUcRDxwGPNFUG2e31ljLS+e+e/fkgLpKIaPETQfbva/TqZtpM9lwHkcxAsbGJ1z0Utxodl3Vpts2eXlte0YRJToXtArhrRQPCNRt/pCANVL5r1vN+1/qmiSHGuRAQyfaggVwcwrPoc+DH8zaKeAwgx/4TUTTqtZt3qA9ENaJcAt7YcIA+COwzSAIUtvP0DhDgmHdZhQzuCCpbMM0/o3MgdGaoZBAsbEi5K89LxsIJxgwsNsYgYG1KmILtbhF3VQCCDM4ghPlMwIP4KYSIuNiGtWoMEp80Qg4lE8iGHCI4+HgCGuE0xGjlUWLJZFLlRCCDIBgQuw8CQQgwE0QdDC/UPHxYjFMmB6X5ccvWSJ+P7BbdYbmsaORBg41EOPf8DhKUoYGFM7SwRwYqZwPhOA2dZhBJkuZJQhmq5ZIpNQo4GCDGhwBjsCBECEW/mA2T87gc8q6ZbQkuaxlJouSdsJDEGSQAzmssjkM2QAJSmMEGuyIWc3kER/4ABpxcqYU5SzN876lzv8l00qgeEMNamCEIOqGIWQjZqaAYINfJoudVkKBE6TABM8EdKCycQIVCDpLhlYID16hQRHqWZqiXGBvoQHb4kbRUPbwYaFKYB8fniCbTIiUpHFi2j9tySw8cHMGQphDPzszCkrUIQwM8BEo4BC+dVKqClUIzzgH8tPw+IGoqXCCH0gDzlwxtVZAyZYkjpADGdQgCHEg5EBOIQcbyCACUMgqKOKgAxnw4JjdciqJoqCHgVDBOwNZa1uzEAU0yJU07kQmR+F0/go8qOEHM5ABuYrwhjjgQAYtyMDi3lCEzsVAB3EIJS4ZhQUrwBUQnKGsZTObiihcljOXuA4jBEKKRtikEQwB7UAYcZ1GCCQUpSXIaR8S2tEywiaMQG1oWVvb2+Z2IK09iG0JgtvZqna3we1tcVMBXFII1xPETa1vHMJc50KXtsgdrm99c9zmJje6oj0IJJL7XNWWDLzdza5yG/EbUjzCIJ8QLmi1e97qota8vMUueberWvw+V7uQ4G591dtf6xp3IJBwLSMigYc1DCEIhpXBC2AgAxoA4QhsyMMomptbhgCYv/mF7n1BPF+H0Ne09v0wev37kJI1osUEnm9oJ0Jc/ju1hhSqSQVrUHNjUtgYx6OB5L2CHK9TKAVjpwgCCyQQgxlU04ic2k53DmKA8HDnrU5QgZSXqrulPTNeeWiAC2xQBPFdLK0nahCDZvcJa+rVzRi0ACUs0eY319nOJjtgAu98KEdW7Mye8nK3THCBvSztz6EacrwOrTaV5gqFD2Faovc86S2SYAOUxvSe8eopKQ6Ca5tOFiPoHMkuN7pWIohYVAzdNPWZTNLttBgnOWPqZb0607fO3CFxbShQ34rWyTpFrw91iI/hS9jb4/KqbxboZXmgap3R4ZMluLRfO3NhwsznQI6N7F1324MY0Ka372TrbJGbWaf4hMICIznP/oRi1Pdi3836vLBFM0pl9IIOaIbn56Zt21PvFneB/oO9Twfc4Jgr3sH9lez2VbtPXniAaKA4MnMrk4hdrniiKCCl0Oyb3goH+ci6sKaQl9zkPMrzyfPI8JExO1T2A7K/KTVvGEa61N/qhKviJHNG1VvlPxeSsIBuJp4zqui/6taj7zptZbvt5tpS1208rjCfmy6lV8+WHKmIUmoP3eunTvXXs+RwX3f56DySYyeX3nWWS7vh2QKBzHAzcYxVHeldZrqyiqb2tS9b7H8/FAeeDXgrZVxZhs+Vyw2FT91EG2OIJ9nFl032LG3gormZOqkJP6Gzg7wMDGD35kU/etKE/o70TRJ100d271rFjuClobuZOXVOP8yPFHzww4L+IE6hArntF1O8nShgPN5k/l52l1BAqYCazih/oQqgQvRP31CIT585fHDCSQdi0lTMhhQLtT6JAH6nP3E8/MI5ahR6f9SkBgIFTHCCZ+UleZPFLVdqCo7x6eUIQwXiCljgLLZKBbfijLgawCwIBCrIBEBAgHQbjVYbGcrTms7bkJQDDgkMFQikkz/4qVPYrM7CrMriLPnjrD3AmRc7CNhaMdAqitVKMQFjBBFTsRAzMOkaMeU6LxWkwf2ywRkksRv8rgDzruvyQRwkQgH7rtHSwYuQreiiriH8LfbSwXT7ix8s/sIkDC+BgC8o7EEk1K0X5EITIwhBcCVM+MIuDEMYM0IwTC8sLEL9ckE0bMMjPLEcSz03REIROwXh2rA78bF00zHXgA3Z0MAUybuWwzu2Y5Qo8w5SoDKksrKBMA8EiI0tUz2M6Z2b0T9F85Q08wxPzLGtocA7wUBAS0S/O79UVMWKgbxbaUVScZ+bKUWlob/+eTpXG8U6QT5eO0SMeb1VtBVgFMZhJMZiNMZjRMZkVMZl/DnaA43duxdnBI1ACIRoxD0G2QOlYh50whdupBfcwx5SyMaKUT7m4wwnYIIqUIDxCxUUaILl+wxSQAARjBeDAr/USKgf0xbnK6iEMsdv/rFHfFGAfPzFaPEo7UsFQFCC7Ys3g2wChBwI+LuCbxyp7mMfP4iC8BDAbNGD7LNIzsA+iMwW7psNzsDIoEoYKgAqzuq9uYqCKPiDeGE/bRyIK7BJeuwWlSwqznCfUkCBVzSTmeQMnUQqmtQW6StKzvgEB1QAoDQTQMCCiYwCEzzAoUSAP/gDBXBAZgGE/zNAKqgrjSCoK8BJrozKrwxL01CCldSWr3yrAGyrtGxLqiTAznA/ttQWDqyCUvhA+auCSqxLZtFLsrQszmACJogCFEABqoyWP1DJvuwMNFCCavwWyKxJEQTBysRMyhyILFCCR1iYQCAoHsuxVPCDhUwF/iZwyi8RTdcozUy4hEuIvq3UFh9bjVT4A9SMl9YkTdYYTX2szR3DsZRQyIthxClrKxRQgkr8luMkBfDgDJu8F+ekMio4AAXATrmMFueEzuOkF+ocwOvMzoUBRc5gx1spz4tJT/L0kUwoyGhZT2aUz/mkz/q0z/vEz/zUz/3kz/70z/8E0AAV0AEl0AI10ANF0ARV0AVl0AZ10AeF0AiV0Aml0Aq10AvF0AzV0A3l0A710A+9UCxQAgiJgvQA0ROdkE8oANTQAwEgQRSFUf3gAwHIggJgzhjFUfl4AgA4gBz1UfmQAgDQzR8lUuoIBACQAgGYyCJlUuUwgIWMggFYas0mRVEnGAAHvE4q1dIt5dIu9dIvBdMwFdMxJdMyNdMzRdM0VdM1ZdM2ddM3hdM4ldM5pdM6tdM7xdM81dM95dM+9dM/BdRAFdRBJdRCNdRDRdREVdRFZdRGddRHhdRIldRJpdRKtVRkDAgAIfkEAGQAAAAsAAAAABcCegGHVkpVWExXWU1YXVNdZVpkZVpkbGBrdmt1eW54dWlzfXN9/6kK/6wT/64b86kT/7Af5Kgbz6ox6qsg/7Ik/7Qr9rMs/7cy/7g2/7o8s61MhbJ6/71E/79K/8BM/8NV/8Zd/8lm/8xu/893/9J/hn2FiH6HiX+IcbSPVbu4eMa9PLnDI7zcFL3rGb3mDb7yAb//BcD/C8L/E8T/G8b/Icf/JMj/LMr/M8z/PM7/Uc7tQc7+SM/+WMz3RdD/S9L/VNT/W9b/ac30YNf/Zdn/bNr/dNz/fd7/cNf8jIOLkIiPk4qSmI+XnZWcmJCXo5yjp6Cmq6Srrqius62zt7G2u7a7vrm+wLq/jcSmtcuhr9S/0s2Q6caN8cqG/9WI/9iQ/9ya99mZ6s2g/96h/+Gq/+Sz/+e6/+i+9OKywr3AndPugd7+jdv4idT0qNzzut7xheD/iuL/k+T/m+b/oef/pOj/rOr/quP5tOz/u+7/s+H1x8LGyMPHysbKzsrO0c3R1dLV2tbZ3drd4dbK/+rE/+3L8uTO/+/R//HV//LbxN3t2trm3d3o1dXj4d/h3ePuzebzwe7+xfD/zPL/1PT/2/b/5OPk5eXu7Ors//Xk//fp//nt7Ozy8O/w4fb+5fj/7Pr/7PX48vL2+Pf3//v09vb59Pz/////9Pn3AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACP4ATQkcSLCgwYMIEypcyLChw4cQI0qcSLGixYsYM2rcyLGjx48gQ4ocSbKkyZMoU6pcybKly5cwY8qcSbOmzZs4c+rcybOnz59AgwodSrSo0aNIkypdyrSp06dQo0qdSrWq1atYs2rdyrWr169gw4odS7as2bNo06pdy7at27dw48qdS7eu3bt48+rdy7ev37+AAwseTLiw4cOIEytezLix466K1EZGG2pT2kqh0i5aW4nU5bSV027KjHayUlF8/BjktEe1wD98Yj+eTZsqCSZPlBAMpYBJFAWZFTiJEoUhacpqj5tVvnwpnyYCSVwa6MeJQCZ6Qul2aPps97Kh0f5iTvvdaBQp112bCpVZFIlKgUgocQKIoWXRoJmTvY+WP1IneggUBRoFBYIEeoFEwckfCOhX24MQDlXFFALRRxAaSARiEIAELbKZKYpUYooli4i4ySKWVbJIZotEFkqJppyYoiKZKbIZKDoksh6MMppSCSMCebijiC9admKNkzFCpCKWxLiij4p4pkiSJqLoZI1AmtLijkYykpmKNX6opJNGPmljkC4ySSaUSAo0ZpFXgpjlli+aqEiKZopZZZVffjjleiGuSaJlZ5ryJiNl9imlizxaCaaWYjrCZYxe+ujnZqQ0imeYAgX6YpNHsgmpKaDwIGmPJIZpGSOSwtljJf40jmrokkbGCiuagFaZ4px+2pkolJ0yqqulbYI4bKiFbqnisEQFksR6wJky3R9IXFicKdItVB5GR8xASVLbihWKiFvRcMdC45XGFBNIIGDFegaYEsUBCtRLIBJKKPCEcR8R4S1SDo4VMFVFALHQwGEhPBQnojTEicIj+TtJhHpJEoNnFCvlX0dGyDBxUeSexV5XNMyh0MZlhaxuSB1/PFS4CaN8lRE/KJSud2thzLLHREGccUmTwKDzz2q1TDRdNph8dFEwcwSHDJII1fRXmzTZ1Rs+JHSzWR9+VtIbUAc1tVfhcUUJDJ8gZInPXI19GBwxRL30WzfEMTdbT0dyd/5bcPQwF9uQDU0S3HLz5DZXVX/VCQydHLR1WYd3ZTVKcOvNk8rgAR4VDnYbJPNYnz8WRwyQ7K1WHDiY/pPmHMkRAx47sY7VyF99AsO3Bck++1qRe+Q67Dn1nlXZXvXwhkGPkyW8Vrpv9HtOoScclhw3GNS8VdHT9rzqZNmOO/c5Yb7SHDGca5P4Y5GS/VXGF7SJ4GRNvvJL5Ndh0/JXjSsW9QUlPxb+WIEfS+oQA/vR5Hrgs4j33oLAxNwhBkpLYPGOJ8Hz0eSBEYQJAKtCvK/wbyCj8ZpmavJAOchkg1TpoFcWKBD/iQWFjIlEDDpXQa30AA41ZIsMaZhDq3ywh/4axIkMceiSri1HfmBh4drIw5mcSEIGFGQJ+sS1vqzogIghFE8Nn6gGIFYlDtXzoksaSJInGoElZJRKGpmywDVCxY1QgSHQZlCElcgxKioEyxV9BMem3FEqVYTJJOiokkBuhXZkAWOM1GLIpVFiBkMQI1Rs1zhJFvInnaBBJE/SSK100ipX7GNTPpmVP5rkkZssiSmdkkewoM6FYTGiyIKSSYOVRJRMwSVSFve9WVryEzb4gQAtSRTOERMlU9wJMH0wzI0kEyyhAAVa1qCDZoLlmbEkyiduIEyRyJIs+jsLJV5QSZwtBTXqGUgoWqPO1EyFFDjAgTWP2ZOk0dMgt/55wrPU2ZvfZOY2UdhOVOCJg7TdcyhvqNlBBcKH7WSLofsyRRP08JzoTEchq1wJKXpwA4N2JKO5RCJZ8nAxtHyzKFJAjymYoJ5QcCI6jUhpetCFFFL8oKMeAelSWhmWSszAfGbRKUucsAcBvYsgBkIPh+RFIKrYlAbSXOhPCibVCVWoPgOxAhIaIRCrmsJCA7mVsQBVqy9N5k9jDcWdYjSeQnlqrZuwlZ8m4whaxchWZ7WrWjPDAxp8C60wUmtZgZVWuMo1WJ2ya1zNmliyAqpPiC3sYFXUWFfFarGWqqxhGZtWKwn2rpwN7GYzC6IP1dWxmxiPWN862Q+tVrF41f5sayPLWtBmVoatWuteCfta1B5Wso/1kaRAlFffQha4ccVTC4v7Wcz2Vq1N2m2zdMObzEyrWgNxFrQUJlSWAOFfGukuUjZBSqtgZgbAg9y6kKCAd3EiXvOqlwIIxC53MQSbQvFXLy2CX6+UtyqWGUIqydJfoFyiYQxhGFeKMAOXSTUnd5jBgz2iy5cY7SIVBtg8yWYKUsTAcuDkHVTABmKKiPcoPAXLEoFwRvUyMioktsh/OQgaz9SBBmaZcQ+fVroJ24QUMHCwj2U8ldGlNyI6VmOGUTwQH0RRLEmeSpSmUr+JnNgoiRMPaRSpvCEn5IF0kMiSH7w4j0rPywgpIf6aY3IDE66ZIgVOyhAhMlyRRRkqWTQFzQi8lpNGhYsPuXJRQiHSsdzMYhu+iqBV98Q6vpklP310WwZJBEmrZAiVtvRD/DyVR2ZaIZyGZqHFssSBRPh/TdxKLWlKmTs/Jc8CiZtY4ozmTtiAmZouyQ+6mGuGJDoqwOzBMMdMT/j98CvENkqsukJQAS56KIS+DHMWV06vPJsori7KRnHqvvzU2CBtDku277ltM/eaI1g798nEQgof2KDa485lsqF9kEmUlGoiHssQwKsl0MR7Kf6L9FeuPZR/G4UIPFtPckCDEExDU90OKULYIJ6RG1MceWeBGx5oPTyDJ+VzQN6vVv44rpU6l2V0aQANya/iQmNa++IOqTLMKZLumbNFzTaPiCSEZvOVsxwPMSAiOD2OlFIfBL0vl4zSIyEDRwuM6EfpTEIcDpmcO+SJn7b6Qiw+83kfBZVaV3hCQj5zggclSwKhBA1sKWppK8TlW0E7WkY964Is89dTgTqWF/IGv3HF5zPfprDDjpCg4d3LejeKg+DJba4gsjkLQfoh8z2/gtj03V1J8VdgKRAgOL2Ua/G6UKIHBL9mnuELmUMYPckU1PABIa9/TWpiHxgGCznsnXiBuRMIUIEK5BJRwK4pFECciK4bPwuBW49nl3hsN8SerFcKH5hgUYIsAQnY1U6glf7OkNEB1Sqa9wrnBVKEASt6KTJdKe0FYiD2y4cJWFWI6IPCNjrEIMy99tmpmbcUohq1QPuUIJcACA1CEOTVQpZBaNI0LpmRWgJhCfdRCQk4Hg44IuQigetBgRr4gBGYgCFUgVmGgaFwCUaygT7SgesBgWwlECA4gRPYgOSigie4HptABy+wBlZTgTIogiZYgSK4RFXTMD44gQtogpYQgx6YgEB4gSjIgDGSg0zYYePhhDpoNVPYg+RygJUBgzDIgQiYgV1ogV+4hWC4gj5ihS+YgSwYhXyUgg3oCDFACZaAhmVIhReFgTNIhRdIh+MiTSCYhU1ogpXAh1cYhkf4gP6Z4YREQQUU8lXxx37CNxD+p06kEQqecTPHUYmVqE6cKBCb6Imd2GGeWIkYk4mjKBCk8IliJ3apCIquuIqa+IqTQQqlGIsKBwku0GJiRzum6Iq06Ik6o4rCGIoY04qwSIyneIw0mIDD+Iq2aIzBmIy82InHYYy9qIy7WIqdGI2oiIqayIzA6Iyc84vYaIvT6IrK0YyiYI7NKIrrUYrFGIqk0YpT9opDoV3VJS3ZhV16cC0PlRChBhbC80RHwEF0d00+owYKhRVmJxL19S6hEC+QKBCigC/6ohiDBASH92gy1EMKZhzzJxa21k1WJ2vq1pA/EZAJsU3yhEcH+RVGt/4QTpYVKrl53DcR5fZGzVdwbBMHfnd+hNcR7YZ5M2d46haSq4MR33V7SYGUPXF4kgd+lGdOF9ExJdaUOykU40cQP6CLVYGSP5GVo6cRRiZvyOcQcpA6VyGWM0c+GXRuRpl/L7YRDyR0AIN6DyEDR5Z3UxlUHcF0n1cUB6hlELFrQImXvsQRGdmUiNkQPpk/QWkStpY1cMlzuQZ4VYGZCvEJ8bR7q8OWYck6emlefXaTHrFRRCkU0UaYhclrVAGWPuGUYfFdIrdmfROZtTcDhfNo9tZrmjkVNYkRHbN8sQmaPhGTDvFhVRGcXSF1JmUSo2NAxSlCEOEDdhkVzImbHf5Rl5KWUNopGEyXdV6GBzKQa9mpFXIHNGu3kSuRZWexlQZBdlORnmbxm1JhnxHRCTcweDthnD1RXjfwlk+Bn99JEZzZeD5GBOK5ZrLJE075A6aXEw2qE5Y4EaqXQqXJRC0hcVc5E+HXnLpzNuwpNmvhn//5Et6HEyYaOxURlXhWoDJBPm72YDMpaSvanzEBdF4ZE4+XORQBB5QZFTe6E7BpODKBdTThnvXZPEwnZaG3cDKhdgtJbrejRjBaE/pZUAsVoJJGoANaE+0WoWM0pDiBnBAxBIHZFF76FEVKpDeBcMS5EuH0ntcTBzogFW16pQiRosdkMXq6GATEQ5JUpf5vlqc5cZ4pEZ5oRKY3AZ8JYQPS6RSICnoamhNSOqIYxqg2AWsSgaZRYagbkRoIVhC0FwruJFVZ6pkV9JiEQRwDQS0SAVD7NBDAJ3wksARRMKsJMaEU+pQ+QANMKRK8ChV++kYzMQUDQBpJQAIR0VDVNxBKgH0MtQTPmhCgehPXuhAI16Eh8aF/h0Dy6RSgygkCECCmMABH9RDpx1IFgl3rmk4HoambChR8ShI9GmIXAakvOhNI8Cx8kKwRsVRRkK4T+VXmOrAGeB8kyIKJGELhMZhZ5LA5eB8OW7GeSLEU27ANe7EsmLE02IAYy7AfS4MdW7Ixkhnqs7ExErInS/6yLquFFQuyJhuxGruyMxsKNrgGKguxpBGzL+uxLfuwLGuxPyuyQTuxN+uyQluy4dG0Q7uzTyuyS2uzKTuyU6uFPFBHVwu0Hxi1ZOi0Sfu1Q3sfIwO2NtuyqSibejAAopAEuuoQXgVW+9hVjegEGqJOHkteKnu1Ncu3Utu1YUu2gHu2ROu3QUuxF8Wzdei1Puu3Zqu4YsuxhJsZ5Flpjlu4jFuzkNu4mVu0h9u3nUuGinu4kru5UCuzLbtInptFRrADShu6Ufu5hOuxj0u7PguxRuKzLSEKA7C26+cQgUBd0XJRptB+xfss+QhqpvkTlGAD/AkSq0mnF5EHOMamNf6hBAZQABTxkPAyt9eBBAdAsAghrzUxrBLxCT2Qmh9hvjZBRrmnqkxGE34AANZBEQfWEMT7aN+1m9wzcYNRCQCwpkRGnUMBN9+3EfcKOmSEAzPKFAJsEaLABMw6E9laExUsEQR0nRkxpzlGRp4qrjMBwArAVX/6EExnfnfDqoKRvyXsEM3bA/CbMXggYWj2wEsxqTiBvuprEdFrFmYqEYuDqTqBw4rGvjVBxDdBCvvGvxThrVvxwxIhA9x6FEjcwjTxBuWzNwxsxYzpFHMAAxqcMR+cS33pYk4RCTOgkRXhxCOXRnAwpUpxwTRhw0pBxzCRpdUGEWycFXdWBzbgFP52zMUqQQr7SzTFOmRGXL5UgcUH7BCJ7KEaAWQxTG/La8ZSQUA7ahzkG8JrJMV+VKJzSRWSkMaHt8ky4UZcyhSmLMgYgccRAgSuOWGrHBOz/BGFDJLflhFGwHYaU8ZdlhVYHKnjG8hG4agMkZafDKXIoRUE9GRy4UYz3BSPzModMcok6Rhn42XEXMxdwZk2UJsuexacWhFAlsdR58uo5hUds5ee+JLi50YxEKxEIcfU/BHk48yJYQONXM97w3RqvBhb7GNV7KRhoZ838D17jBXGzBBAgM9F8QVbkGpoYXJgAQSjGc4dvBFFsKBDQQgW8ABhwM9a4ToOLRjeaRSZEP4CCxACo/qd0+wS/izEckEHakkUXrAAG3AI9kiVPD0WlHADN/AIbqcReFC9QmEGFDABZEAQ9CwT2yyYaEHIF51jMl0QTRoUmPABCzACLe0jIu0V9pfJftGbPyEKI7AAHoAJc1HLL8HWYzTKuIavGjFOP0EGD2ABg5AQbu0STR1ElCEin/Cr8rwVC70Qn6B7PIEIG8AAXqAtT9qY4DEQRhADDYxsHUFOOqEJIrAAIJAJ/PLVYXEHMjAEVY0W8ZwTYtAAGGAIevHUzkcZzNG8N2DOngRH/ksTHv0AXwARrk0UA42noCFSSjzV38oRNMDOL4EJKs3SEdHXEeF61nOqpv7gB7Pna8oMF6NT0kgjzC/RBQvQAYgAFibgG76HLbi6T8RHHF0N2qYgCTTQA1GFF/oaE0hNAUsNFtNXrRWFLZegfQ7R21IT3Jv5Ay66lnCkzzCR1QvQBesNEb8tE+tKe+nnBHwQH0rABHeLUZX8dMf8AmI9FZZQ2gQx3yxh1gvwAZqAEQ8eEwLbVAYrIGgggASYjuqkjc4ojd5IjeYoj9IYj7Bo4+Uojfqx4zeOju1oi7vI4+vRjrWI40XOi5+oM9YI5L14jgUBCe9NCfNY5RT5jKrIjVZejdRI5Tm+5EW+i0SujOR4jaSgz7RjjdtIime+5UJuCmTQAHiN5nOu4/7UaBSMeFUDEbePuFRrOIY/aF1seOhm2ILLWIZDaOiJSIFW2CSkIOlhiIc8aCT9HSMXpYKElohAiIJ5iIGphboiqIU9aIXToYeegbuQvuhWs4ORfumBCIOtHohEaNFt8IBIWIagkIVRWBklSOuvXjWGmIZ66IWjHoY6GIGzboaHuOxspc+PLu06GOwU+IG97iOFsNhcEIaXEOxL9OmwnoFp2CzIO7zHu139WK0IQdGWDB6NRNLmBUfmkhKaoNIgkOIc4dwQwb2hoL0rxV7vUpH5YnxaoxYADhR77N46QNuj1BEIbhKiMAYMsAGE8BELPxIfWRAdbwoPw94Q8QkEjv7cajHxJJHbYtAYL82j1w0Ro8PRO9URMxCnH6HgItDgCIzOLyQa/zUJN2ADg13McHTaIuEFDADeJOHvLrHXLOH07VmVM/QUcEQKmA0SZZDU900SUC/yMzHDPgDxImmZHYEIHrDgENL1KaH2KJFGgS0D+wzVGxE0HmHiH6DWKMH2KLHi1ttqG2F/Mj8Uha0QaNwRdp3nKsH0LdHyMMH4LwFHk2ADQt/FGuHHG0EIGMDYi+/1eFEEQVcWyIwR+b7S/P4zG/8Tp98TCd0QWI4D4OwT41wRRoDCEiEGF8/aL5H6PqH4duRvIEEKnh/GPjH4CfED2t0QHt0AKx8TvM/5O/4xw64PFiQeEQo+As5fbEMw9V5h9BFx00qvOnwPwmjx6yUB9mJ/E1AsESIaEWUwARNQBtgq0T09dCbxCRZd2ToR+xMRzQ9xCBzAAADRxdRAggUNHkSYUOHCgZUYPoQYUeJEihUtXsSYUeNGjh09fmRYR0YPSiBNmoTjI6KoLgs+YDoZU+ZMmqZC1cSZU+dOjaJ4fioSwwhPogZ7wIE45oGFQUUp3nQaleIiqVUhKrKaNWEoS0Uj2bARSWvCSlA3xpDEkBAGBl/GMsT6Vq6pTXPfOrSbNZRZnqSMwBhCKm9djpJiLNQUYkEITXkREnYcWfJkypUVTsIxo45li3B6KP4Uw2DDIc6lnUI2jRN1app8ncaJAaSTXNcYfSA9OIjChDGpV7OeSRV4zbjDZXLN2ulHjDd3a1ckhdYgpg8LRvhMXdz48e0zf3f3+NwpHhs07mj9XlGODYMtO8AEnh78fPr1S8sZOQk8DtymyDygoAz7BtSoKwI5uuTAjfaa6xMiYiBCsKjki4iUSWCYDREOFvACPLwU1Eg4EDHSbsSnKJRKEhxkmCOqsi7qBI84fMDhlBFcagy8Ek18iseLxPORISCzkmOGG/DgTJQ4aHDhBRSWIiRIKaekskqFSIEjhh7SoqyTHWCQYQUIsJDQSio/NFMhNNM8KBQUtQIqhh9Kyv5pkyEVoiQHDVw4QajZ6nOETYZEFPSgHQu16c2xOhkihiH+pMmSOxESBQwHUGhSBjnKnI9QRD8FFcRJfIihCEjHMmO3LGCwAQcj9As1VllnnUsSUomg0yrqrBPFhiEmoYRTWu07FFFPP3UTuEmAgMEHLkF68SGWFnjPlB9wGPFYUNcEtVhBkx2OEiJg6EGsj+x8iIwJKGjKFDhmOJVAb4elt97KOgmqhzmElekQDhjocKA6YjDX3gMnNRNhKxWm7BM4aJChiFwxAlKUGz/IhKAsNzOR4SrnZRPkhBUtDY8fYLhhU4yiNUiMBjaIkiAiZIDER5FL44MPhHImSBSeH/4iucqgp2RwwE7emGEGIpCsqDZCLGjALYJGpWHiEYeejAQmnlDCIK25HkjrKLo2uGwe8QBCBhmGOE8iTjNRLATsTAEKsE/MfosPspFIcCA+mgj7kr8DZwhrKQ0P0uPh8JhZBiDmsNqgTySpA49gvVhgNII6gSM/KhHPSwopBmJC54FEJ90P1E1hwg+4aL1ZaAN9xKOIG2CY4Yc4nh2IFDlogAGGFCqYgIyBOokDBxh04JjK2ClzQo+BorCCoOint8IJNKbffiHFpfwe79RIwUONHmJ4gQYdhngjjttZOCGDK94YIrMXcIgjXqLnq2KKgZwABEH69z9ADNAUACxIJf7iogiHhGIRddnEIm6iwIEoIi6LcAgoFAFBCZqCgqZgoE02SBdFTHCBF2zgCDdRQg+iUIQQZOEHLTgQDL6QhCakYQphiEMQ4qWGoVBhDBeowxu2sIIuBOIOjdhDmzhCiSusoHBCmMQifnARWAlFJbpCRQXycIpBnKAUiUhFKIJwiDZcIQ8dgUQwLvGHbZRhXByRwUU84g5rCAIQbBCDGKygBS6QgQ+KwIY8fAKIJrzJGitIxDR6UIxoFOIiJUnGDlbikWSMoXAWoUlGRtKMFazLIYETiCTYRAE3SRApTRkKVYbilAzh1qdiWShSgE5BZ4BABFQQPJV1iz5MQAICqv4XCgOQLpjVY50JhPmQ5zlvVuAq1CE2wAAt+IAGSoOVL+nDibkVhJve7Kb4xEklTYhgMY2Z3CT4NU52ttOdTgkNBmL2TjZpC1HNBJ8twTMICzygNwZhWajwOaVZFmqgPoLmlDIBgsWEsyG0Kig9JTrRmmCuA4igaEY1OpnwGSdVxWNnR222ToM+U5+mQYQHrAORgGpzVrOLVUTTVLQRWcwl8AEaRDe6U55e5D9M6SmyaHVSBYlCpJU5RAcYIJCnHBVERAXRQW32TJjWJzENtQi6ZCVVhA6VVk51TDzn2TSvBtWsZ33aA8RwVlnJNE1uXRhY37IrETi0IqCwa6H6Nv4re5ZUVqCAa2Xcg1GOtPRTXOWRXA9EUrbipAy7EZBH8looxTbWsmZK6QKYetlZBfZjJgWOTTF2Eq3GdFiIzdYzPSuXn7brJIa9J2dlW6+1AGy2t8VtrK7KmNyeFnagrcwXREManMC2UH0t1GoJClzJ8LMBa9VJaQXaW+pOaVfXqW69Kjug7dqnuyCxKGF58l36kJc+qI0qc+XyWAoYzynGFRR6n1pWWUEVJIjoAIeqYl8C8Te7/9WKaHEKYHv51z4GLq95LaIUoOqFvlv9rawSGpXaSk0rkpqVfBWk4O5weDsefshuJ+sUEA+nxARG8UdCswFDpHicypUSjBGK4P6I6IY3jpHutoaF3HqqNyfXHbFV4BsyF3M0uxYdcJGVvGTH/CdATG6nIhhL5L/KuCKImOZmKYNhCM/HZ34IZyj4AOaB/CFnpluIlXmkYe7SuCCizZFlhpwmNmdFbKUsyJ0HooAo9BnKoXJZg/8cJD4wgXB+A5wpSHCJUJBt0KYJMkYIcYGoPXpKq3MCmlfXukCQQAlMCODr+EpVk2TCnHIbzpzNxAjjBIIKVTjgHrBnPemZgnqBiMIlAIEATjCkqqByc4e7K1zNGSfY3fl1af4gOlFQwX8HDLUpDIhA68maIJbMoU0eWEUpf5KJmLyJIi4JRymeEdxLnOK2G4ltb/5/UYky7KQX2fhuc8PxhJO09yTVzcII8vCN9KYhFrfNCUX02hIvgjckvdjAOZqCE4Jo7yU86W5uHxHfAP92GyVu8YxjnOLrvnfHKy7yRpZxhiSfOCM7KHHBJPzc7A4hKUYoStYEomuuRKUpWnlKPUQhbMkuyIlZI/TUEL00Ri/NsYkCzGUS05jLNAUSlIAAn4u6y7FCTluRbpk6Z+USdv0mQRgNka1bRunGKTtl0q52S7fd7QBWs4niDiKahursqd5xhLF+d9ao2kpd9+5XaRXpt89k7YVHfOIVv3jGN97xj4d85CWv0S8jxMxmqjxCAhEIzI/5IJzYg+uqJGY0E/6E9Gwac5hDXyg9F4QJSpCCAgg/IBIsIQp4DjoCnl2lrznalEyIwiunJDbfKxr4uKdS74OuAODLPk16O7QpAIEEv+2V0EuI/kA+TQUrDU7Re/WDE0hX6yDpwdDfJ0ihs0/oRC+aIOEnXempFIXRHRDN2XOCE/5QpU2LfiBUAMDdmxL6Sx3TuwlRIIG5S43+IwgCZB3/kxI/e0CC4IReMwUFUMDUKCDucwJriwJYmx4EAIQ/UAAL5BFAeLXrsbXu8aCucbYp2UAVjAIWHIhAQIL6kxIZRKYDqrUZrJIOnLWC6DQclJJl87lpizYpeALuCZJlk4IXhDaCUAIlcAISIP4Ba/ORP6A/JCwINEACzqMSLvy/Z6O2KZk2MBwIK0CCRhAUm1slU0iQP6A+U1ACNpwSN8Q5OITDSqiEPjNBKdm5nJNDM8HDV0qlmxO+OyylPEyQ6fsUpqseTigmU5ACJLhBK4FEmyiAggBAM8lEp4uCA1CAUaRBH/nETWSdY/JEVQRFUSRFQQk7gwiF2TuQWAwVW0QUTniOr2MTXJy8XwTGYBTGYSTGYjTGY0TGZFTGZWTGZnTGZ4TGaJTGaaTGarTGa8TGbNTGbeTGbvTGbwTHcBTHcSTHcjTHc0THdFTHdWTHdnTHd4TH+qgCJIAKJ1jCeMRH0+CEAiglPRCAaKTLx4CkDD4QACsogKoTyIScjCYAgANQyIeUjCcAgDmEyIp8i0AAgCcQAO6zyI60CgOgPicYgAz0yJKsCCYYAAsURZNkyZZ0yZeEyZiUyZmkyZq0yZvEyZzUyZ3kyZ70yZ8EyqAUyqEkyqI0yqNEyqRUyqVkyqZ0yqeEyqiUyqmkyqq0yqvEyqzUyq3kyq70yq8Ey7AUy7Eky7I0y7NEy7RUy50MCAAh+QQAZAAAACwAAAAAFwJ6AYdWSlVYTFdZTVhdU11lWmRlWmRsYGt2a3V5bnh1aXN9c33/pgT/qQv/rBT/rhr/sB/lpxvXrDL8riD/sSL/tCr/tzP/ujuLsnamtGu2um7guEv/vUT1tkX/wEz/w1X/xl3wwl3/yWb/zG7/z3f/0n+GfYWIfoeJf4hws49ltZpLuLVauKpmu6o9ucIlvNsdveITvewBv/48wdsFwP8Lwv8awfQTxP8bxv8WwvUyxOohx/8kyP8syv8zzP88zv9Bzv5Iz/5YzfVF0P9L0v9U1P9b1v9pzfRg1/9l2f9s2v903P993v9w1/yMg4uQiI+TipKYj5edlZyYkJeRtoGjnKOnoKarpKuuqK6zrbO3sba7tru+ub7Aur/Vy4r/1Yj/04H/2JH/25n10ZL/3qH/4ar/5LL/57vCvcCN2umd0+6B3v6N2/iJ1PSo3PO63vGF4P+K4v+T5P+b5v+h5/+k6P+s6v+q4/m07P+87v+z4fXHwsbIw8fKxsrOys7RzdHV0tXa1tnd2t3h1sr/6sP/7cz46dD/8NT/893E3e3a2ubd3ejV1ePh3+Hd4+7N5vPB7v7F8P/L8v/U9P/c9v/k4+Tl5e7s6uz/9uT/9+j/+O3s7PLh6/Tw7/Dh9v7q9Prl+P/s+v/y8vb49/f/+/T29vn0/P////8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI/gBNCRxIsKDBgwgTKlzIsKHDhxAjSpxIsaLFixgzatzIsaPHjyBDihxJsqTJkyhTqlzJsqXLlzBjypxJs6bNmzhz6tzJs6fPn0CDCh1KtKjRo0iTKl3KtKnTp1CjSp1KtarVq1izat3KtavXr2DDih1LtqzZs2jTql3Ltq3bt3Djyp1Lt67du3jz6t3Lt6/fv4ADCx5MuLDhw4gTK17MuLHjronURkYbSlNaSqTSKlqL+XJaUpbRasqMdrJSUXz8GOS0R7XAP3xiP55Nm2qJKFWeEAylIMoVBaFMKaBy5QpD0miRn1VelnnZ4Er5SBFYwtJAP1QERtETSrdD02fB/pfVFPps59JLr2DR7tpUqOCiSlAKVOIJlUAMy5/V39z52NFp8WcUFXoIdAUXBQXSxHqBXMHJHwdAV9uEFB61RRYCUQEIQWc0gV9BBBKkyGamJEKJKZQkUokpmihiWSKTKRIZKYqc2KJlKWaWyGae/ICIKTTa6CKKMW4WJItDapKIjpPBCGSNSAaXY4lN2piIZZUowqRAMj5pmZI4amlKlyUK+eWSJZLY5ZE3EimQk6SY6F6SigTn5JhGQgkmimLeKWcocrYoJYlwBnolkZaN+GZkgH5ZJ59bsjlkisGRqUgjc1o56Jt5mglpmm+eSOOKbU6pqCdBYBpKkonYOSSc/oqQOimaipbp3qFKShnjjHrOuiiQf8YapZtUAuspJX1WiaSjrnIZGbKeEhWIbrwFZ90fTXBonCnVLSReRkzcMElS345Fyopb6VDHQueFx1QUTSCAYCgGmHLFAQrke4YpTTyhQBXHfZSEuEhJuBxXSxSxkH/mNsWJKA1xYrBKStggSYV6RRIDwxgPJeBGS1hs1InJfWyVDnMoBCBaJKdVLsg2RFLUy2FV1tUSRCjUrlk0GxbyxUNx3DFKkswg9NBj/Yw0XTykvHRRJI4ER8xC9fyVJuhy9cYQCe1cVtTolfQG1UBZ7RVoXk0yAygIeU2W2YfBQYPMT7/VQxx1szU1/iR5B/wVHELMdTRYcHMkN908Fb7VuV91MkMnB7k9luJcVTJ4R3LzzVPL+4HlA94Grdw5xnHQ8EjfasXhA+o/TXySHDTgsZPrZNGeFSgzjFvQ5V3ZPnlLsMueE+VZMQ6WEG8YJLlYxGvle0nB52RyWKTwPpUcPaym1vS1Rc86Wbjr/n1OnLM0Bw133FT+eGIhX5An1muVddgvnb9uTc1fFcr8XmFf0PJhyR9W4ieSOtDgfjMhYPHEEj6CKHCAqLsDDZw2vq+4r4I3ESBIJEhBmGiQKmgTi/8GIjqzrO9rNZGgHGTywanYjIG5GwgACTc+SNAAdBjcihDgkMO22BCH/j28ygiDGJMWhsSGPHQJ2J7Dv680EEUPtIoRp3JCmUTCBslrSRWpx72t/CCJJSzLFvN2RTUQ0SpxyN4ZX/K8mFxxCSxp41eiGJUG0rEqcvzKEm8iiRsoYSVThAopxsiVL0LRZWvpokz6+MeUKLIrpPCEWdLonjtO5ZF5m8QNkLDGOsawkyvBZEw0ycmTiFIr1TvLF/PYlVNuJZAmIeVJYOmU/Z1FdZSQJFr22JygeEIHCisJK7liSag4TpeUAaVAQMEDIhRTmTO5GzRRQkicMHMIz9TZZ5BZFjj4IJtPqaYeiQKKHgROJLwUyyDRMokYcLMstIQJatozkFC0pp6p/pkKKXrgA7ZN0yhN+6dBblMFJ+ymN78Jzm2u4J2okEII3xQoUXAm0YHwwTvdEggfAGYKKehBOtSxjkLiqZKH9sCfHSHpUl54FjfQAJxNSedQsLAeU0ShPaHgBHUYQVP2sAsppBgCD1C6EZUqxXjmURf9kEKFPRgIQQRR0HpCZK99gZAIPHhnRXeihGBK9EIZ2tBAuNAERggErKbQEEFSFKpbnUlXv7JVo1iEph21NU5v5VNcG3GiuSopM2y1FY0sA6jgBEEH47oTlPyKpsDKCa903VSxBIurxioLsoXV62QfW1nAEkpUrArOXzXL2TMBdjJsHaxbTZGlzDAiRqKq/qxk7WoKvq52tI6NbV6RJZDc3tayd5UtaZXFWM+awoaYgixuUUvczpJWt7eSEqY2C13cfhZYZwpTb5trWmL9SUXRLcq03AMcU1wrWwMJhEGr5S2mFIFgGjFqUpBqQlLcQHjwfFcTFIAgTtTrXvlSwL7gJS+GiBMoAxPfRQ68OFdiZTRISIJ5mmIJiDHkYVypGNC2qpM72IDDHhkmUJR2EREXD6ZNyQwpaKC52q1Fvi/Z20VgfBT6ijEzRYBjfrcXlbG1eCIOhiBlMlMHHZglyJ2c2ulAbBNSzGDDTL4Ikn9SOvxGZMogNDFWJDSELP7nxSi2if0mQmOjYO0ypFHd/o6jfBAOSiTMbDaFJ9ZGFjhXUIVxjgkPVphnijAYKUiESJmLYmOyWG4gS+DaWP68FZk+pYwPGTRR1sky5ETipb/rM0Ku2EhNq+QG6fP0WvooYVGjJMKmfoijoyLLhaz6bE1ctHLucINMe2Yrv/QqQqZ7FpYeWTkrRtxXGB3lTvAAm6kmCRHMmOyFeYWZQmCYnbfqnCHOEcxn80FEDSLpodiy0gVxHORoyOM5CuGkofsMll3onB7w+SuxbvZATEpUeWtka/ZGyLqXElQejFsg+3botAt2EElg+movJgsS4DumtPhajLQDNbkDVJYkiMw9nxl4jRGC6q9o2dQVE3a+/itS5JH/7yxywwOxrxLJAPnHyQreysqxwuuylC4NDp95VShhu899pdvQnMMMEGhyieC76GzBM9IjEgmjIV3nVUlEoJmYlkMj5L5eAfpQtB4Usz6CBp1WZ8ChAsCOc4Xr07xiqZfekJIXXeNLaTXbhQbzotecZwSZBDDF8m3zPM/nZ+eMWsp3TbjzJISjS8gbzilztkeknNF2fEKKZvgKjh0qtCOFD9ANycoHjSFYx3XCEXkQUmD136iMd1h4vpAihB0raAfKx6/CvSIgFpIOZ8gc1KiVy8eEFLI5CB8GAhs/DF8wGpZ8uOkcxIU2VCCWuAJ6hVMcjqqs3AqR25Kz/uL7pjwyoL1fCh+iEFKCQKEJ6O1OpD1ftYaULtS07z5TWL8QJZQyK7H/SE9tevz0opc+TxAFH6IQ7CcUvDNmQnYwDEFri7MUVHUgCWJQptAglgAIEUIQmkAylGAZgyRJnsB6GSgQlcCBueQenZGBmRGCKEKCkoSCLEIyI9hbJCha6KKClLAig/QlJ6iBGjiDJpiCPQgknREKOyiCQUiEdBADa1CDQchahLWDQCiDQhgcRJgZlYAuG9hbkkSEogWDGoiFwfGBUbiCQuiDZ8ZaQZiDP2heMIguMThIY1gJaQiCRUiGZFiFLziGWYgiLViHN6iFa+iCNoiDWQiHLNII/jQwCX9IhoYoiHO4hVDohFLYgabACX4YGlmIhy4oh5NYgprohYCIh0SBVmoVVdM3EE3lQKSRSu2yiq4oEKkEJPMGi6s4b68oi0BCGqFQi7kIi9ARi8hxi8DIixjXi8YYi5MxjLA4i48AA42kjMc4i8dIjL94ixISjLSYjbgYi9tIjKtIHsZ4jdK4jeLIjcLojdoIjdg4jQKxi8vYjumIjrk4GupIi58DjPUkjerIi8EojPH4j7gYkP4Ij+yIJu8oLetVXiI1geilB9uSUQmRfz1BPFfEBFVRaLLWEGqQM/inXwcwL/UiEAoiEKLQL/+iGH1UBAWIQRpDRBjGEO/B/hjG5kyONzf2JpE88WoIUU7b9hR9V18O0WVZoZNeAXVUoUH0JkiqBxZhlBBxwHhSpHwe0W+ol2+UJ28rCRTF9F5Qxm+fARGhZxVZWTaSkREh82PzJX9LMUMFQQQ6FpXYl3gXUWVMgXhm4Y4OIQerQ3tSKRLn00HNdpXJNntW4XsSlETz9ZUQYQNWdkmjt1QZAQk28HpE8YFoBhHLdhU4yROEiUcekZKJmRzx85T605clYWyKlmyCaWpGSUUgAQraVm+yp5ZK0ZQLYQPb55pqQZSBBxIP5W9E8ZMQFxFDwGxHuRZjKRbvFXN9Rpqm+RdKcAMil2eXlmx3h0IkETK5/tkTnLCUV6NALFYVvMkVrSkV42kRpUN0h0ebScGWBzEEiGmez7kSBhSfcfYGHClqyWkWaqdpePBhpraZOqF1kgBM++kQ3XmZEVF3UyGgOuGdw6YSnWBOB3ocWvUfD9QDgBlO8+kSsMl5UZYEa6dpnUkV7MkQRHB7OMGNd0kRu2eia3GeHdkSFYOWMyGcZOGeB6E2FToSDio9cbkS73cTLPocUcSYjtmhMXE+77ZVROBlfXaiSCGlDoEHNPCWMlGkZDFlRxcVVHoUP5pBVmQDIwoTKmgeUSSZDboWJapPM6F3+fk0dMSggqSkNTGh/VRRGupp5RkVfboRQaWiLmGX/uNBR0hAmfP3mO5iExa3nSrhCRDaFTqKEHHwA1IRpnZKEEOadjSQqYthQEAESp+UZ5haEzKKEpJZpidBaWhqETygnkxxqrBXljoBpytJqIZ2R4caFaV6EanhH/0XCvm0VXgqmz3knINRHMR3ig6xUBI4ENE3fSUABVfwrAnRozmBrQszBIJqEtraZBdRnXUqE1kwANDhBCUQERdVfgPxBOinUVDArhFJqz1hcTYaEg+Xo3dEp03xo5wgAAViCgMAVQ+xfzeVIOhlsPR0EF9aFA2rEZtKEqTQpl5qSTxAB1DxsA3RBAbFB+YaEQ9IsCKJXiGLgaFhCaGhCcEBGpnx/kLgyCIrCyDf9rLgCBox244pm7Ix27Kh4bI6CyQAQrM/y7LukbMAN7Qya7QskrIqlrRHC7Q3u7RPC7MAx7MpqLQqm4Rr0LRXC3DQobJA27VSS7VE67NTC7YhZLZUS7Uzi7VRq7Y167RFe7Y7O7djC7Zki7VGa7N2K7RAUgQWGQoi5bd5K7Z+y7dpu7doK7eEu7g4S7dV6xJ6MACi4ATW6hCkKFb+d1YYklYD2Leitbg8C7lEG0I0+7Vce7dIa7Vji7JeC7SPe7eo27eDO7RDqLdhC7uq+7pli7ul27Nuawr/+Uemu7q627h1Wxmhy7tJO7qym7twy4EAwrdwW7fI/gu9wEu6jBu0X/u2wWsKSwAElZi9z9u7kMsiqXu6kVu82su6fkseddsSojAAk9t/DzFe7LWQIzmBCdlGsgqXQaF3PzBwOLpoJZoHRvYUmPoEBlAAFEFgILm52tEEH5kfavGtEAEKQpBVIaGlDXMRnbAxPlkTfgAA2UERFdYQC6lpRUA2FWQD9+oXlAAAf4oRGisUNbwScgN/HNFyolFMPtCka0kTohAF6ToTvYrESFGfU5nD81dMu+oUPzrDCmBWnvoQknl/qIOsgbHCV+wQk8ADQmCsSIMHtRZnThyrS6HBwJkRBSwWVmcRjoPBGfG/UkTHM2HHTbZw0zkRGCkW/iWIETDsFHr8xTPxBujTN0FsyKHZFEJnn0MTxXWpqHj3FJBwAypZEX+8es8EB3GqFEk8E2k8xMbET1X5ELg6Fl5McjzgFKPMyCxBCu/Vx7UhrlFGsVGByzqByDyMyoqJEU5GxkWhy1ARykVUFQaEpQ2RrxmZEYPMFMYcEzccFNPMEpGAyfFTzT6hy3vKFNoMyyQxoSBaG0VgnCD2zT2Bzisxyw0xsS6nEQnjfZS8ZleByLBqEG/MyRqhl/3Kphe8FQYEpQiBx1m6EWacYuBMFddMk4+hNmz2yu3ZFbDJA8wJi+qsE7bpx4+TqPS6qDeDpPgcqeQJTja5FNGc0Bxx/j4CnRg80JgojTqSydCKschMVsiXKhad4AMUXU8iLXPgVAQrbRQn/RIQjRTXCUlIANKpDMfgpASqahRH/dLXQwOQPBj4GWUEXdBl8QjYnNUwAVNzsJdHNc9vcxaT0AM94AgKmhF4kMCgvBY9vRVxPUBJ7dJikdET8QgAqhRFLdU7cT7K/BcGd8tB+hzXjGx1xhHt5M1kbWtmsU4arANdOWzgBAoxIMztpxbE7BRe/dWIRgNCfDYwFQOnPMx+LRYepsV8QQOTLVBznRWvzXLMEcY9cKHcB1MujBR9fRQ2zasBEmtFEDvDRsxhiRRD/RCoYb/tOKymYHzGdxedjROl/lPVdfGqQ3MCvvF83EKtEqgA1XfaB3HJQGDbcmHdGDN+8gpS3GIJ6vcdu/kZS2l6xX0VcZwR5p0UvR0TBtt/+0cFfACAAsgQ+a3A8M0QchADgU0Vk+oQPNDLRjHgL/GAVpVWAXsFZ0CBFmgwsSiO7BiNxtiNAHmOByni4aiPJG6LAImXJI6P71iPLb7iJp6OMe7hAhniIT6QpvAIOvADk1CNB8nhLg7iHb7iAwmNHC7kLI7k8lg9Db6PBOnkxajkIz7j16hiRH6QQ6EFnVuKppC5BEFVADeHJKiHmDjmefiCndiyl1iGJtiFZD6FZ66CnIgihGjmNoiJktSIoMiI/lnogpbJWnUO53I+h26O5naYiXU454VICTolhoZ+hWnOIocG6XzOgi3cBkY4iZXQ6Guuho7YW1ho5yRz50LI6XF4hJsuZ52+6GG456xugg1O6pKo55nO5zT45mo4SIVO6pRoiH/+h7lOf0OhXuRlLfxb7A4prwhx3CzB7CUV17AT1OEEUzrg4DPjkfPiwDa1XwhCCiZpfV0zeOrGdDte2t4HUw1OykvxkgXB7pW42Z4KCkQw32xR7UwW3YP6zxNROk/NbzB1A45q2h1dyfsxZZLQAzzQ2u0JU6xt0olU2Ft6EUtwQyMMqKS9FBcN3kRhxmOsFnPMZBkPpLmHEaAw/gQ2YO3DDE5Fw9jvTXq9FtsIcT5JgO8NseAMccnQ7M8jnxwccfA9oPBBAVN10MpeqfF4kQRUXRb8zGS7XRRNHzRqCQk64AMV/RN4LRFLoNpH8fRE4ewq4fUokc8XQQpKkPREYfML8aQ5b/R7YcZUDxb3zfb5RgpIQPFe0fA1PfB1BvMRYcZDYO44gfYJwaNrf2uQWWcNCwotHNrSk00HzRRWLPd+UQc2IARVHxVwkJpJMQiGgJwdWoCgUPYJHhU7dBSiMAhfsAEMwABeEKN6by58bxGQwAM8EMMyIewXUdJBcfpf0AELwAAb4AWEUCIPb/iPrc2ksAQzgAQEffUP/mHLPcH7G7AADtABXzD8BBH7kv8WkuADN3DPSwE4PCEKZkAC0+8AHhAGg1AXIY8T7X8T8C4RpVMEgB/LGgGfOEH+I2AB1J/+nW/BAGFK4ECCBQ0eRJhQ4UKGDR2aUvRQ4kSKFS2aSnRR40aOD0lV6sixExEab0KGpESKIykakU52zGRmRIUFDjyEMTQx40uePStq8hlUqEKgQ40eDXX0IR4eOu4obRhK5UY5PKAuvFRm5oIJH8Iculj06liyZc2eRZtWqBwbQiSpPekDDtpLZETQnBBiDFi4fY2C9BuYoCXBhUklLQwqCY0kU8+KvThpBiiydUNQ4Kr3Ek9KhT1H/vQceGdouB9JR/JhYw7alBuXDIFqGTOFEGQyCR1NurTuvo55o/XtWc6NHnh+IyRlo87QQ2FCTFhQQYTt49WtX8ee/SwpODSEuMT+pofP5h8eRJ9+W/v6h53ZK3X/figpyLpBKaFBZNJVTYgrgqLBuJDKO6+CEcoQBa1G5FMKNAZxe1CoUACrrhMkaECiE/iCkygJ2DYiJAwPHGDAghHMSBAuByNksUUXX0xIkiFoUEJD0iSh4a2KCPFixAVMRBFGIYckskizIpkxif0M40GJiQYJowMSNyAhSCOJzO1KilbUUqJQ6sNOkiJmGAI8zjhsiAgf0DSIxw2m/GIQ67js/lKi+OqUKEs8GfqSwUmSmEEISHiqhE2F4LjBxoNEGcSLDRZgYIMvCNFOzz0vxTTTkzrBT4g5DD2qDhoGLUgUM77YgAEHOvCCUk2v9O/VhGKV9SBQswMFDh1sUGJJjW41hRRPOlGpu+UEMpWEDmq6Sc4IgZXV0loHknbaPl/Eg4gZepADWoRaS2iSN4QQIgkibHhElDJIsIBZnGCslj0++Dho3oFEsZchMKcdaF9+DxuykzduuCEJASei1aA5bpghhhlw6IImm94l0l/2SoiiiicKKkEKjQXC+IqN+SW5ZIbuQMIGG5B4yiNDQUFDhRZmSCECEL4yuTo+Rm6CMIH4/pACZE6ABtnnhCyeFmlZAdYSj5RtKGIOXw8CJZI78JjEN0zs0uACGFxQ4YUhpr5SaeywwEKgKOgVCG2BqPDDbVOi8GOheGu9+1XT8MRDiR5muIGIOMwUiBQ5dJghUEgyIWMmCKaQIYYeEAf8DW9ZzPs6KvQQ6AouBtq8cy6oOEP0hRIuGXWSL3+RFDzUEIKGGXTwAYk34vgbhxZQwOACFlrAIQYf4gAFFDzggKMOsq9k/bctsngbkIGej556U6iQfiBKdkqkM1IUAUoTRZLaXqBEdlKkM0/AN0V8lcrHyPtEwk/kfe7R7yyU+duv3xT40w/W/jSRCPLdTyAA1B/9/ux3QPkp0H/ccw8CBdg/+HUvgPQroPnwd0H+LTB+HBxgUgZoPtBYMIEdfOABM0KK/Z2wEjupYANRSIkSyvCEI8QIBEFIwBSaohEbvCEFN8hCDPZQEaBphPoU4Qg8sMEIQcgBDF4wRRjUYAhKYEMeQCE+EW7CEx9Z0AddSEHQAJCI/MvgB8UowPH5r4Y7JF8Zy5i/CXrwfBzUn+qqEwgnmCIUCkgKYfjoR0AO8o96FMidcqbIktFnWocgQwgKlIEVwCAGNOjWIl0UhSYg4HOhMIDamnCAz83tBJ5kSOY0pUpMacJsRCJQdEZAnToIQQc6QIKOTMZK63AiRQXxJUE4/tG8nBVTVoYQkQNkaSWCVC1rxoRmNKV5KWT6CEi/nGY2DUKnkvFyT9cSEo88wIBIVQmbCwFXMb15KUZ2E5qOdBGPOkDODXjBWRTRBDHr1E5t9tOfoXETpOp5z38W1KDT1GdfGOWFZa1qUgetSELr1D9jrrNOe7NOstrVAA+0KijpzJlF8UShYvJzWvDkjalG0K4H3MRVQ8knNE0KUZrW9CEq3ehNcmJTnkYlmq/UlER9writdAVnZiGFUI0E1FVGU6Ra0gRJz7K1uyzgAV7hS1pAuktpIrJWTL1UUs9Sl6rmZQybCYxXZQXWnrbVZLJZAG2o41a6XmSmtbqr3jhx/pTmPCeutVEPb74ITaMZk5ske+qVPJFXixxiDJJEz1yts1V3RlOpRrpsP2NpIASxJ7NE+mxdRbue5vjIQMwcbWpPwtimGhOlEwmRNU90TgbF1JisbW1FoamJwjJEnFMiQRmIRFnEqta4EZInPat0XOY2162E+MI8I2VP51YXQtBMrJEmdK9BoEpV9XwpO0MLo8OSDLeYOu+eSGEJMyirJqwKr6Zsq07r1rcsyXqUTTxqX/4ixbL8islKreoBMcS3kdFUa7Sc+qpMaEViN+ELOBeZ4Exlt2w/vVRWiuoVtBaErXj6MIj7O2KKkBUvIQhDh0m84rKEuE4utg5c8yLZ/oUwrZgUzhSMu2ThIvG4MJaBjlwD+5Dt3na8L/IxaP8bob5CRzo0jug7l8zi+pZHmZwdMpW1HJj07qnLcCntlQ9E25XoeKlHdtGX8VTeabFZMNVU5jWHImGTEZdfSd7yVdD8kt8ugEqozXOgBc2e5JbTDING9HEoSt++QFe6kiKoWRYr0z1jzkX48kNwQsEHP6ToD/Nim0LUvGO0MCq6732oX4pc0kpHCM9wCVkfCRKykSngCrdO9KLam98OpDrXvw4KH6JQtIEQzRQlsEQoRgZsU+CXWftldrR7IjcqhFpudAtECZ4Qhewp5NUvcrNGcOqunVaHFKO+kp2n9W2y/gRCC1u4HudM4TnQydtzgbiCJQBxgL0qRKomM7NBVCoxrK7ntQCPZm+18we0kcJ62Jse9K7X7evtgSA0ZKAf2RfCB6rkjmt0YCLeeMYBvq+EOgyiHfO3cR5iPIfmo6MDYwjHHn7cjHV0I8z9yAEHfEAMgoijx1GO85l/r4juM6IMS27EFbIcKC9MpA5J3nKpJ9EUnCC61CeYxo+bMOs6n3ogDQjyGaJvQaFgOdeHXsSib51/GtR5ymuu9DZaQohxJ3oZV9hCHrLHkArYqyA3dkg9XAFk/y4IjjGl+LBKebebHOUnCyBKBZSyCU9AgOHttmBjTvq2jMcTu9MSTIOQ/l4gyfbp46EJ+jqxvvXShn3so43udL/T9VCNJu2vFO5Xif5ZiOcXJW5/Jd+zqNXPiiaZZV+W4S/f+c+HfvSlP33qV9/618d+9heC6YP4gQ/Nlw/3DxKIQLxK/MLcQ90ytelQD4T9teL0OUORfn6FrAkFicITsqAATZUACleQNWE6AC14FYz5GPdTgCi4AkDCFFrjGAUMwAbMmGUjJAVUAOWrk50RiBMwmkC4P1PwA4XTEj6AAmIjiG0jwEwxNmQbCD+gArWRNzzRg2E7NqMRNhO8lBU0GhdUm/bDlCtIm+sJtTOgAirgtky5NvUTCC1gQon7wSCkG/dLClEoAd3T/o4kHAggVBsl3BMtnBsl5IR+UwAr1A5AYMKKM53OOYA/+APAwxMz3ILQmbfSSaSN0QInrBMz1AI5vAI6HAgPDMI94UM/HMRM4cNS+sMSCMQ9Ybi0ebhuwwLNo7c6acQ7jJ6BeIInoIITKAGLw5NGfESCOIMmKD9MCcUllDiIM8VULEWB4IImYIRpCYTBY0DC8IMPfAIyzI5ZJKRAMgVLoARKuLV+u5S/88U/+MBM4cVD+kVTWEYGxBRjbEZASMZp4SRS8qNQmrcSaALNwxROQqVQmLyBOENNucZPCqUrOAAFYEc/xJNz9KNxBEdE/MbIy8Z5W8d2nBbTIwgMrBN+/jQZgOQXgfxFf+wSgtS+hFTIhWTIhnTIh4TIiJTIiaTIirTIi8TIjNTIjeTIjvTIjwTJkBTJkSTJkjTJk0TJlFTJlWTJlnTJl4TJmJTJmaTJmrTJm8TJnNRJ7duCJkAMKqiCnRRKBuGEAugjPRAAihvKpcQOPhAALigAb2TKqbwOKQCAA6DKrLyOKgCAatTKr/SMQACAKhCAFATLswwMA7g/KhgAXUTLt7SIKBiAfltHuLTLu8TLvNTLveTLvvTLvwTMwBTMwSTMwjTMw0TMxFTMxWTMxnTMx4TMyJTMyaTMyrTMy8TMzNTMzeTMzvTMzwTN0BTN0STN0jTN00TNF9RUzdVkzdZ0zdeEzdiUzdmkzdoskoAAACH5BABkAAAALAAAAAAXAnoBh1ZKVVhMV1lNWF1TXWVaZGVaZGxga3ZrdXlueHVpc31zff+mAv+pCv+rEf+uGf+xIv+0Kv+3M/+6O4Cyf9W2Tv+9RP/ATP/DVf/GXf/JZv/Mbv/Pd//Sf4Z9hYh+h4l/iF++t3K+tj25wje7zCK83ii92yy70za80hy94xW96gy+9AG//iK95Fi/wwXA/wvC/xPE/xvG/yHH/yTI/yzK/zPM/zzO/0bE2m7JzHzJw0HO/kjP/ljM90XQ/0vS/1TU/1vW/2DX/2XZ/2za/3Tc/33e/4yDi5CIj5OKkpiPl52VnJiQl6Oco6egpqukq66orrOts7extru2u765vsC6v67Hl4rErJLFs//ViP/SgP/YkP/bmejRrf/eofXfqv/hqv/ksv/nu8K9wLrYwZ3T7oHe/o3b+Kjc87re8YXg/4ri/5Pk/5vm/6Hn/6To/6zq/6rj+bTs/7zu/8fCxsjDx8rGys7Kzs/bzNHN0dXS1drW2d3a3d3W0ODSw//qw//tzPXjxv/v0v/w1P/z3cTd7dra5t3d6NXV4+Hf4d3j7sHu/s/k8sXw/8zy/9T0/9z2/+Tj5OXl7uzq7Onk4f/25f/57uzs8vDv8OH2/ur0+uX4/+z6//Ly9vj39//79Pb2+fT8/////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAj+AEMJHEiwoMGDCBMqXMiwocOHECNKnEixosWLGDNq3Mixo8ePIEOKHEmypMmTKFOqXMmypcuXMGPKnEmzps2bOHPq3Mmzp8+fQIMKHUq0qNGjSJMqXcq0qdOnUKNKnUq1qtWrWLNq3cq1q9evYMOKHUu2rNmzaNOqXcu2rdu3cOPKnUu3rt27ePPq3cu3r9+/gAMLHky4sOHDiBMrXsy4seOuhdRGRsvJUlpIn9IaWov5ctpPltFayox2stJOdfAYvERHtcA8dWI/nk2bagclTZAQ5KRAyRMFnEIpYPLkCUPSaJGfVV6WedngSussEdhB0kA8TAQqmcNJt0PTZ8H+l7UU+mzn0kufQNHuOhSn4J06QNrTAQmTPQzLn9Xf3PnY0WnxZxQTcwj0BBUF7WHEens8cUkeB0BX24QUHjVFFAIxoQdBYhiBX0EEEmTIZqEUAkkokBQSSSiWGGJZIZMZEtknhpzYomUpBlfIZpnoQEgoNNroIooxbhYki0NaUkhmMArU5JEtBpcik5OZyGIhlkViCJUCyQhkkliiuGUoXpYopGVKMkmil1AOmaKTM1rJSZKG6FjklzaGCcmYTZrpnpVRigknkIDqGeaIg3ISZqB7ctkmjoUEV6YhibhX45VSkrgjnkjiyKemJ9K44o1EZoZoJjxUOieakZY45JP+hozq5pJkgvonq1LGOOOlpL5ZYpwnzilrroN+AqibfFZ5JpI6rhnZnssStYduvAVnXR5GcGhcKNUtJF5GQ8TwSFLfjvXJilvJ8MZC54XHlBJGIIAgJwaE8sQBCuQrRihGIKFAE8d9FO64R0m4HFdFALGQf+Y2dUknDV1isErhOlKhXo28wPDFQwm4UREwWFzUicl5bJUMbSgEIFokp1XuxzA0UtTLYZ3bVRE/KNSuWTQbBrLIQm3MMUqOuCD00GP9jDRdNKS8dFEkjqRGzEL1/BV5XqXhQ0I7lxU1eiWlQTVQVntVmVePuLAJQl2TVfZharwg89Nv1bAG3WxNzQj+3gF/pUYPcx0N1tscxT03T4RvxQm6XWHiAiYHtT1W4lxFInhHce/NU8tmfTLxVjbcbdDK+3G8xguK8K3WGjao/tPnJrHxghw7wT6W7Vht4gLBBF3eFe5iUd6R7LTnJDxWi4fVQxoGSR78WsCTRHxOJtfs+1Rs1LCaWtXXNr3rZOnOO/g4cc5SGy/EcZP5ZH2SiVjLF5TJ9VoxDvZL6K9b0/FXccI+V9kriPPCwj+s0E8kb3iB/mZyQPJpRHy9U0sDGxOHFzjNgV6JHwZtUkCQVPCCMOkgVUAzlgAOhHRm+R9ZvhaTCrJBJiKcytnEAkGBDHBw5GPEC0S3Qa30QA3+PWSLDnkYRKuYsIghxIkOgegSFpIleWOp4Q2/EkOqqDAmjYAB81pyRbCQkCw6YCIKy9JFumWxDEisyhq0l0aXRA8mWSwCS97olQlKBYJ2tAodveJEmzgiBkRYSRWj8okybiWMKMojVQYple7N5I+BTIkj6/i+sqzRPYqcyiT59ogYCKGNUKkhKCX5k05+8iSb5MonMgmVMO6RK6mEDFBMeRJGPgWKlrQBJCp5lj62LyiZkIHCSvLKrbDyKY7j5VmKybdN0OAHxxylTOwmTZQY8ibO9EE0ufYZZZJFDTbYplOuyUeibKIG0BSJL2tGTq08YgXedBtTUNOegXCiNfb+TM1UPlGDcFbzKE37p0Fu04Qj7KY3vwnObZ7gHUL2wAZrE+hQcCbRgdTBO90SSB0AFoolzEE61LGOQmzJkk/0gAYR7QhJmzLDs6BBY2hZp1CgsJ5QKKE9nLgEdRBBU/awCymf+AFKPbJSptiMZeq6H1KYQAcDIYggClpPiOy1rxEKNZ4V1QkRhinRC2VoQwOhghEQIRCvhkJDBPGVnBbVKrUq61ZXUpOTQsXWTA0qFIkIFlszo9ZQDUlRweGBDMbVp0spilV8fauxcCWoX82VUIgl0mMX+ye7OtZPlE1TY61EI8Zqdk+T3atkJdtZuGopM4iIEV0ZC1pXCSSvcP3+rGLr2ti+xpZWts1sWzX11sPGVVA6rJRuE2vD3oq2tZwNE2BRVKnLJjeyyF3tlTxV3Mf6VrbWVVFlizIt9wAnFNfK1kD2YNBqeYspQBDXRoq6lKOa5xMxKF5Z2NsSeCkAQZeo173ypYB9wUteDGmnTwamEQFn5Yv7+YQQhmCepkgCYgx5GFeIEAOgZTUncYDBhT3CTJ8o7SIdvsoqPwOkF2juiWuhL0zEdmKKqBio9iNjZoAgx/muJZZCYbFFcKwVcbI0M2+QgVl4PMqppW7DNvmECyyM5B1P5XTyjQiRRRxiPQ7EB1v8T4p9bJP8TeTFR7FEjMlyHtbZuMkI+aD+RLiM5lBkQm2/bPNBXChnmNDghXWeiIGNskSIgNko7jWL5QZShK2NZc9ZkalTzviQPxelkJdBTsbYTBRHuy6LkcyzSmKgPk2v5Y8M9jRKFizqhygaKrRcyKm7EmgZEyQOMZgcZ7gSTK4ipLnLnLJVUPgJuYkF0RvGBA20WWqS/ACNxV6YV5zZA4ZROqvOOeJXnq0UWrHaBv40iKWHgssUMsdxkMMh97zYgxqk9ISf0XVVOOGcGuD5K2NOdu/KfW55a0Rr9kaIupvyCR/QINwC2fc+qV2wgzgCpldLMVmAMNiBrJorLU0h7Dgt7gCVZQghE0iVqzLi5CCE1F/ZeJ7+iTC2fF8kyCYX4FniJgdgcxyrYhmjQJQ8vq243Cq4tuQLyJAW/13GdqGjYsob0gYXLHDoEcE30tdC56VDpGgED+LNF9nn58RbLINGCMVlKRm1kFURL8i0WBDs7YSAnCvblmgWQ+10hrwh1kOPulJS7XSh0XzoOecZQR4hTLF0m4zACzraZ+0ZgmRT7jsh+3gWonSbtx0i52z24xEC9ZQLXIYH+YQNzF1HxA9FcPGltcJdhpCg/pvVVw8LJKIHBLFjJe3cHndC0ltzA/acIW1go1YuT5NPyOYgdRgIbPAQfMFQmMmPd1y9HbjQhgpEEk8Qr3CKw1GVyV4hZYDBkbP+wnuobDKgu19KHZQQUoIkwQji7U6jPV+1hpyu01fBGsveSIRTZgX2IumpTYs/XvHSBwlK8CEKwX5B4xDo4wYHRmIMAWvGtBRTdSAJYlCh0CCSoAcRQhCWQDKQYBmF9D6ZsHosQjKRwIG75B6dkYGZkYE2RILvg4IhKBAjuIJAAoJiFnAauCKFhCYnqIEaSILB4T8p2IMzmBlA+IKhEAlCWEhusAJmgC4qiCKhEYNF6IJPuIEz+IOdEQnoYoUo8j7+ExxPiIQ2tIXB8YFBKIM5eIUsgi5iCIUzaBlFKAkiyIYkeIZHmIQ0uIMy6IZTqIdu2IUhaIeQsIVeuIOCiIP+VlhIKZgIL/AIg4iGhmiEiViCfZgZMeiGHRgKl+CHXGiFlXiHkFiIdtiGlDhFOmFWaAVV0jcQTNU7pDFi7fKKsjhzrzgQswgktniLt8huM0eLMwcdHYccuqiL9uSLQHKLkxGMxKgIKlBjnmOMwWiL0FiLGjeN1diL2KiM2SiM0CiNQII1HSch3JiN4niM3qiNuLiN1piO6YiOvMiOwEiM2zga6CiLoRON19iOw5iNudiN5viP9eiN+oiN7mGN1saOQ0Fe3mUt/ScQc7AtGZUQ+Ec2EbF2HJd6YGGKoVAGOXN/72IEBzAv9SIQCiIQndAv/6IYf2RrSNcIK1BEEsb+EO/BGMKWTk7na/I2kT/xcAZxTtl2Sxj5FRoZCliWaISnVBFhUpz3FK1GFjKHEGsAOK83eR7Rb6c3dJWXbAQYFKyUXsiXFFv5E/QTeiI2ekhJESDTYmDZfU8xlKHwAzVmFTr5E2z5FN0HZUZVlz/2EGzQOvFHlSKBPiCUbFlZaiJHFXVZQUwElgr4EDAQZZpklu7CEYwAA65HFB8YaRBxbFcxlz5xmJjXESvJmMlBP1HZP4BZEsJmaMVWmJ42dVMBmwixCdi2fK+jl0zxlAqhfVYhm1LBk1uhYiZ1lULxd2TGTD6AbIu0FmE5FrRXaqeZmn9BYYeTZxlTbHnnNSX+ATLb5xOXEJReoZsKYWJVAZyOpxkmcTpHxxOKR2YN5AOL+ZvSuRKKqWlp0JGe1pxnUZls12ZyoGGi5pmIkxKOIEz6+RDfqZlJuTvLqRbg6RW+CRGYUAOSpxPuEyATVAOD2Zbz6RK0uZRNNgT9WWeg2Ugt8QMNhxMdt0wUkXuIuRbm6ZEtQXJqORPGeWgTlDYHWhICyhO4mZsv8X43saJm8Y4S8ZiR2aExgT7vllU/kGV19qNLIaURIQcvEJcyQaSLRxGN532SqXdYBAMj+hJP+F4UUZlT0aO1I0EzwXf4+TR5dHeEpKQ1MaEQVVEaqmkRyqG95wMp6hLt+R95JAT+l8kUe/oUamo8N4Fx3bkSmfCgXOGWBrEGOiAViUqnk5o+/3SdmKoYCUREoMSgdXapNxGjKMGfJXWohqpINLCeTGGqU0l6OuGmKxGoWKdIhBoVpJoRqeEf/McJ+pRVEwqiSBSdg1EcwreKDrFQEjgQ0Cd9HZAET9CsCbGjidcT/SYDXykS1poTmcSpTDkTUTAA0HEEHRARF1V+A4EE6KdRSaCuEtl1AwYDNRoSEedqFSGnTnGplyAABRIKA/BUD6F/N5Ug4kWw9XQQVJoUC+sRQloSzygamUQDCGiXNGEEBlUH5BoRDyiwJCleHYuBoSEJoWEJwQEaRFiyJXuyAAL+RfIHjib7jRqnsgHHsik7swG3st+YgjTLIjbrHj37sj87Qy9bspmBskCbs9/IskrLIjrLIinLs01rsktoBkcbtNARs0hbtDW7s0nrtFPLtF8rtF3rslgrs2NLs0g7Q0Srs2t7tj6LtlwLtqDxg0HLgTwQSJcAIHOrtXx7t0uLtm0buG87tW4bGoMbt1DrEnMwAJ1wBNTqEKgIVg0ZCqgogGMLhn57s2Abt0j7RWS7uFt7tm8rteB1uDjbuTG7uEQrUmSriGmrtJ8LuJ7bsoA7u6lLtv8ZSEf1ugACuqRru5rbtaUbu7XLuVw7u3Ybtl7bt0OLuMGLvHybtWKbuKv+WwQ78LXW+7zMK7p/27Xea7jEa7umizXk8bMt0QkD0Lj89xDdZV4iNYH+V17fpRCwKpdDwXc6QG03+muHGQdChqg1gQQGUAAU8V8iWbk2BZIeq29sOhSbcFIAV5XdqqIYgQkrUMEdUaIcgQcAkB0U8WANEb95ll7VCT4lJxiQAACqWhENexQtjBJpoKlVCXNhIZ4PYQNNuhQxTBGdoATnOhO7uj9I8alV2cNIIakfV6hIcakrrABk1akPUZn2pzrGGhgkLMUO8Qg00AO2iTRyAHdohsRIcb82EcHECWKQuhVZZxGOo8EbYcZVcZBmIcc2YcIZ0ZSHZsMQQa9OYcf+WiwTMwx/T6PDgUyaTVF08Yk0ucpvX3pmTsEIMQAEcbrGWqHECKEGb1ptR9lgUGGnEwwRthoWWVwRb0AD43TIRoXHQwOuSMbBUAHLOTHIEwHHNRFNSvbFRSHLAiyrU5FAWNoQ9+qeGuHHTDHENfHCRqHMWDTJ9MPMsacReQqkqoxMNUCstAEEyrlh0NwxWcHKC8PLuRlNCdMU3TwUyCzEWjHDrmoQ/at60dSXTZHONvrAWZFAUFp6jYkRYezI1TwVjRADNukYaTPGakHGOkGbNFB7M3fOQIHDovw4hvrI8uQVIAOZGmfJWYHJCYGTnPzPVoE++ZwYNIDRIM03lTn+0IlhyEgGyLoqFphgAwttTxqNFRyNEEAw0kZBzzKB0EaRnawmBEiKtmfRxhhBBGP60yetFbKzyIRxn01my7dsForgzFLNQBzRBn6pFFdNEzydRGbxCNe8CAqaEXIQwB+dFjWNFWvNcUJt0je8TYoAoErh00udE7ITzH9xcE0mztRMGQFNbHGmEe/012dZ0ctxIhGsrcT8QCugy+2nFn7dXvZcBC+ww53HESsQykYx2XctFBlWxXzxAttaTW1tFac9QszBxTXAx7smTikMwzAqr/sRb0AwO0Ipy2RJLvP0e7sRrKFAfMR3F129E6fj1HbRqkPzAb7hfNwirRKoANT+99kHIck74NpvodwXM37wClLcIgnq9x1q4dJMiZFBtdtsLU7anRTkDRMEy3/6xwR18H8ByBDtva+fAZ5ssAJ6bUXqTchHcd8u8YBVdVb/+gRiQIEWaDDheI77OJDqCJDUGJAR7o4Ovo79KOEaN+EPPo4UDuEf3uEXDpACKeIafuLIoQgyoAOPEI8EWY4oPuIUXo/yaOEEaeIgTo36SANxQOPX6OMFWeE67uGkIY5H2+E6PhRSgCFnRbmTSxBTZYOQWIeSGBqTGISCSOV9OOUmCIac6IOBaISkiIhUXoVW/j6wG4Z4qIOZkZlHSOZqqOaQ6OVZLoNbDoqYCAk6ZYb+Yr6FZT5oWsjlHQgEMHAGMJiEkbDnX/6GYV6Ffj6ERmjmM6jooyiEi0PpVc7ohVSGIoiHWMjjkn6EZd7pc76GdZ6Gm97oeIjmneHmj4jqICgt9MuQClktDwmvCPHVL6HrLaHHCSE7Ot2W4iQDAL7THxmS7mHAC3xfQIKS1cdNhbccPNYIK87ZTgHREcHjPOwwEFYQMSkQEkPdkPcDQ/0WxI5kxR0T6Q4TdiQ7Sd1e4hQDjdrZFC1roqFujlADNFDaMCxOpL0UvO4SDv3Q14eWO3RLHPEJmz2l4j4XYezFavHGSDbwdCnZGbEJPgADxd7Z21Q0ht1LtF2kqV0Q6DP+BOv+EDdd3WLM2xZ/e/uMEfleA/wulhxxykbV8HgxBC+A3F4hz0hm10QB9EHzo4wgAzbA0HS5TUUg2rId8mC6TCNfrUSw8yOzTU96zDi/F2F89GCx3llvbwp28F7x7y3t9O0T9Q0Rxj5g7TmR8gWho1h/0GZvLsy8CYSO2dQTTf080V/vF28AAz2A9FGhBqy5YSePNAS4CVPf31HxQ/6Mnr7cOWgvEYxAAzRQrzMR6xjh0WktGnJfMiVVBC4gBF2N7Q3hykox+X2vFo5gAzHQzkvxN21G8T5B+z3h2Q9xOkDA9r2uEfB57bMd+c2h+m78Ay8Q7Czh9jPH+Z1fOhb+5/IxIQc0cO4xYaQVwQaoDPyrnxhsAPgz/xM2wPMVRfxOUcrmgvsUsQk6b/ICjxFpw/tGIfRDIeBxvxzkvxGN4Pp4bxLKX2hQEfAAEUrgQIIFDR5EmPBTQoYNHT6EGFGiwYUTLV7EmJFhRY0dG7KJUUOOR5IWP8V4U1LlSpYtXb6EGVPmTJo1bR78pOZFj0Y3Saah4VPoUKIXIRVFGvFoUqYKLTV1uInIix+PbFrilHHTi5FQWybyGpagIbFiC5X1yikSWoSYhLwQgokmJI4Wh/hg65GSnz55/f4FHFhwS0c+XhCRy9bRC0eDEVL680XLhgsSHCxY4ACLY86dPX/+/tvI8BCrYj/RIDK406AwXbBkqACBwQIGDypg2LAljKBOoG+e9S2TbHCYnJ5+dgTEhY+eLele/GGjrtdBfr68rhDh8oIHEi5s0PLlTyWGw4nDXHreJXD1LI0HfzRkJyOWkaY/VBMjcVHI1ylD2M6BCC7QAAvxKLmIvfYWZLBBBw/CZCob2rjPpjdeoK+mSv4AY4sNMKjgAcwcgMCCDLDoIoxBHmQrKxY9cvFFjSr8bBM1ZIiBiNJmnEinlFqqRJAwtuAANhEzg6CCDDjQQkWaYpQxIwWjtGhKKiN6j0E5fnChBjZofOg5iIaAQZGOVmvtNQsimK2223IDY7yirAz+ro46DrJzoE7ydOi4Ky3y88+IPoFyQUzSiCGGIbqaqFCB7nukBxl2fKi66zS4QDvMuvsuvD8QFCtQ4jpQogkkCupgCVMFIvWJUwWFNVah5AACBhiEiEOi6T7BRA5HOOlNKheE2AShDSWjTIIjGRiwwC/8WFHWpOp41QhJBqpjCVYvyZbVaxkSVdqEwhWXIkcflINMGIBog1KDNmnkDTkeWSinF1xQYQhH1IChh8Y27PDDEEdM8kTdovWN3M+ggEIgJe4UiGGBmMBD4lCUwKMhOsstaGOOH11LUDmIqMGFGH5Yo7mBPmFDBhdc6IE+TO5VIQUUXLihCglko822JXX++6M3Bz3ujIk5BHqCioGMRpoKJsRouqFzPx5oaqpDAVPGT+Qoo4d7ZbBBiDTWqOGFFEoQwQorQBghhRVECIGC77aIkzxBs+Zsiigm1mMgvfn+OxQm+h4IEuAKOeoTQ56yxJCsDBeoEOAMOSqTxUNpfCHIQ0Ec60IYL0Tzwyc/ipPPMQ89lM0p9xz0x0cXiHXTQRc99sRPtyT1zTsPRXbcdT/8dtdVh7134VGvnfPjc88q98iH63x25ImP/axPTpc+EuB2X1536I+X3nnOg28d9dcFSoT0UMIH3vb1L2ee+t7B+iSRtTjh444xrgihhbZJIEEKSCCDG+RABCVwmwv+egKJRCxEcpErHfy8B8HyxY97FAyf41T3vQoW4nHDMQQIS/e75D3wek8xndUAs4cjrE8BWbkWC13ICRly4oUNSc/VCpJDHT5KYQ4CmIcqs6wImMhAOSiBClzwgi91Ig0wWOIQilUuHgZHCUZAgNI4YQCHGeEASrvYB7LoEKJ9rIzlssQPQVOJQAypSAOjTcGw0CSEEeQNkoqBEBqDNUwwwl3SOiNnLiG0ggySIJfAWw8VSSU0uQY2PHMTbnTDG4nAi16LxGQmNake1lzqAgDalHfAIx67bdKUJTEPJgMprSxBpT+T+WSAvOMsP4AKPYkU1yrFVcUe6jJWn1AjTIL+KLBlScCIXwBDHW9iCVxKi5enhGY0NRIkNz5yNiRSEgcmSUhpdtOb31xJMxvSSDXxjDtv2gIpwWkaU6Yuk76E1SdCFhFLYQFTmuLOgEb5Kd+ISZWmnCcmn0k1YBrLOrBUFsEIZCBoOYiZmhzoOiXqoE4IyUMgOhI2T5QiZU7Uo4FR4ceCeSU0aQE75vSZNnfDTWl9QpyxGinH4AmrmfpmNdbBjgTaxCk4AWISEH0pTU0ZUo7FtDOdeCVl8CmghR4IJ0EVFFHR+FGqFmWYQxxREU/0rI5W1at/iujHwuoVahLJSJhhQMG0mUySfOISmswEVP/0LU2mcpE1jQk5rdn+M3TGiaUrycRYdynXK+H1T4S9EmI9Us97boenBpLTVz2i2ChRVrKYTCpWM9MsAznpsp/1qGDFZddpcuiicGSAMQ3mWb8UNJMPzaRoc7lJ0joETWaNzWzOuaQmBeKvgvmEbGPlz7uClmqs2YKadMpXSfrVuM+F7kT7Y89MBUifnrJldLW7XYMYtkHHQuh2mNXUhparlYskriJr20PhqqesxCSYklDEhez28LyKhO0/uSsWveZ2U2/q7W/3O2ACS0Sqf0HTdTKwJt12R5LiEfBBDtxSTVr2RRMGpGr2gqzqZrWp/OyId2WkFohiWKibNKpPAKYFDmjWAapFEWtbkuL+X1r4QTSGaYEjUtY3ZjSlB9NxkIWcFBxjpL/LPWdzgwaVIh/WxFFFcTtZ0knq4pOnniolW0R84YAqMr29NCVhwZusjHL2CzJ2jI0fpOYGsbnNHl0xfDdrxC6wdch3xjNxlnLbN0LSZ3MMg2+p1N67NflK+UWvKbnwgDYVLDcrLdd6qXZf9rp5QVuWESXAEIgs69DSeQZ1qEU9alL7xJ36xWRggapJTHtlT3iYDifqgAeh5cFOEGMIoQXVageROLafVg+vm9KqFhKkVa9SwBOUXWpmN5stdVCCt7ClrVB0QBKceJWztb1tpliMCbi2GMb20AEkKIFwDBF2gyT9seD+rvqdwdmDFKYguKOFImlLq3fS9vAESejhAG9lSJfxW2FDU6ngVKLrZ/LAsE8EbnB+25vgzi04OhAEEiJ83/AuXj0MkpBzHDzh9DY+vo7TTn7Rk+AHI0c+9pWweyrvHcZDnjuXd/B8JFfe+jx+QcUNL3PyY93M29c760nwKdoTCM9/d/PO1e8ol9g5+YTOdJZHnYJTD4Ukihc9q88v4+Y7edU1LvWlo27lJRc52RmnQa3XvOUb5HjIUwiaGirgrTE8lQ05MYcnsErgBXkylCs8eE0evCxX/OL6CtBFBYDRCEhAQN81tsl0L0jVsQ18YRlkSINwXiDXfkjmD71J0Uf+qfSm53bqVb96i29S14k9/YsML6PX/2ndZqzw368GidizqPIMArZ6es+iCLPeJ8M3fvKVv3zmN9/5z4d+9KU/feobedYHwUMdkH/U6x9kD3so16s7T4eMsZJPBJE1rsU1a5ZygvxXa5URCqIEJERBAeLqQBKeUOxDHkAK5SKVVakaBVCCJ7ihWDk2VClA/pOVAMw2FypABSg+KqEWgfgAutoD+QsFPEi4P6mDJJA2gii3/5OWbqk2usIDJnCYehOUOYi2E8SWF7S2EqS2GRSIFHQY9ZOVJ2gYwcE1p2ECc5OWcCs/gZACKYgCEtzBHsSYqsmKTuiA2gMNIhwIHnT+mCKEFSu8mCK8BIBTACkEDT2Ygv9jgoqzNzCyNwTQgzywO0HRA3ljGnuDmqQ7lSSMlTeUgjh8gjkciAzsQVjRQz4MRGnRQzQUiHH7Q1hZuL5zuHODgiaImj9ZOChAQr4ZCCRAAib4gA5gwSuZxEYkCDEwAvCTFVA0woh7uFJERVIUCCowAkSgmj3Iuxu6ljzQQCQAw8+QxRnKuqybBEhQttnTxRbSu160xXLZxWLEO16UlbqDoVDQAw28GsTTIi6ytw4wAsmTlSsaI05YvIE4wnKhxvXhoic4AAVARz4UlHH0xi4aI3Fhx3I8x3SkGs8jiAmkEntUJH28Gn7MOnz+jBJ/rL6BJMiCNMiDRMiEVMiFZMiGdMiHhMiIlMiJpMiKtMiLxMiM1MiN5MiO9MiPBMmQFMmRJMmSNMmTRMmUVMmVZMmWdMmXhMmYlMmZpMmaDIUpMAIXYQJItMme9I1LKIAWmgMBmDifNErHqAMBoIIC0MajdMrBWAIAOICnpMrBaAIAkMaq1Eq02AMAaAIBUMKtFEuoMAD5Y4IByMWxVMuOUIIBALhzXMu4lMu5pMu6tMu7xMu81Mu95Mu+9Mu/BMzAFMzBJMzCNMzDRMzEVMzFZMzGdMzHhMzIlMzJpMzKtMzLxMzM1MzN5MzO9MzPBM3QFM3RJM3SNM3TRM0P1FTN1WTN1nTN14TNFwkIADs="/> <figcaption><div class="markdown"><p>Geometric interpretation of $L=\lim_{x \rightarrow 0} x^2 / (\sqrt{1 + x} - 1 - x^2)$. At $0$ this limit is indeterminate of the form $0/0$. The value for a fixed $x$ can be seen as the slope of a secant line of a parametric plot of the two functions, plotted as $(g, f)$. In this figure, the limiting "tangent" line has $0$ slope, corresponding to the limit $L$. In general, L'Hopital's rule is nothing more than a statement about slopes of tangent lines.</p> </div></figcaption> </figure> </div></p><h2>Generalizations</h2><p>L'Hopital's rule generalizes to other indeterminate forms, in particular $\infty/\infty$ can be proved at the same time as $0/0$ with a more careful <a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule#General_proof">proof</a>.</p><p>In addition, indeterminate forms of the type $0 \cdot \infty$, $0^0$ and $\infty^\infty$ can be re-expressed to be in the form $0/0$ or $\infty/\infty$.</p><p>For example, consider</p>$$~
\lim_{x \rightarrow \infty} \frac{x}{e^x}.
~$$<p>We see it is of the form $\infty/\infty$ (That we are taking a limit at $\infty$ is also a generalization.) We have by the generalized L'Hopital rule that this limit will exist and be equal to this one, should it exist:</p>$$~
\lim_{x \rightarrow \infty} \frac{1}{e^x}.
~$$<p>This limit is, of course, $0$, as it is of the form $1/\infty$. It is not hard to build up from here to show that for any integer value of $n>0$ that:</p>$$~
\lim_{x \rightarrow \infty} \frac{x^n}{e^x} = 0.
~$$<p>This is an expression of the fact that exponential functions grow faster than polynomial functions.</p><h5>Examples</h5><ul>
<li>What is the limit $x \log(x)$ as $x \rightarrow 0+$?</li>
</ul><p>Rewriting, we see this is just:</p>$$~
\lim_{x \rightarrow 0+}\frac{\log(x)}{1/x}.
~$$<p>L'Hopital's rule clearly applies to one sided limits, as well as two (our proof sketch used one-sided limits), so this limit will equal the following, should it exist:</p>$$~
\lim_{x \rightarrow 0+}\frac{1/x}{-1/x^2} = \lim_{x \rightarrow 0+} -x = 0.
~$$<ul>
<li>What is the limit $x^x$ as $x \rightarrow 0+$? The expression is of the form $0^0$, which is indeterminate. (Even though floating point math defines the value as $1$.) We can rewrite this by taking a log:</li>
</ul>$$~
x^x = \exp(\log(x^x)) = \exp(x \log(x)) = \exp(\log(x)/(1/x)).
~$$<p>Be just saw that $\lim_{x \rightarrow 0+}\log(x)/(1/x) = 0$. So by the rules for limits of compositions and the fact that $e^x$ is continuous, we see $\lim_{x \rightarrow 0+} x^x = e^0 = 1$.</p><ul>
<li>L'Hopital himself was interested in this limit for $a > 0$</li>
</ul>$$~
\lim_{x \rightarrow a} \frac{\sqrt{2a^3\cdot x-x^4} - a\cdot(a^2\cdot x)^{1/3}}{ a - (a\cdot x^3)^{1/4}}.
~$$<p>These derivatives can be done by hand, but to avoid any minor mistakes we utilize <code>SymPy</code> taking care to use rational numbers for the fractional powers, so as not to lose precision through floating point roundoff:</p><pre class="sourceCode julia">using SymPy
a,x = symbols("a, x", positive=true, real=true)
f(x) = sqrt(2a^3*x - x^4) - a * (a^2*x)^(1//3)
g(x) = a - (a*x^3)^(1//4)</pre>
<pre class="output">
g (generic function with 1 method)</pre>
<p>We can see that at $x=a$ we have the indeterminate form $0/0$:</p><pre class="sourceCode julia">f(a), g(a)</pre>
<pre class="output">
(0,0)</pre>
<p>What about the derivatives?</p><pre class="sourceCode julia">fp, gp = subs(diff(f(x),x), x, a), subs(diff(g(x),x), x, a)</pre>
<pre class="output">
(-4*a/3,-3/4)</pre>
<p>Their ratio will not be indeterminate, so the limit in question is just the ratio:</p><pre class="sourceCode julia">fp/gp</pre>
<div class="well well-sm">
$$\frac{16 a}{9}$$</div>
<p>Of course, we could have just relied on <code>limit</code>, which knows about L'Hopital's rule:</p><pre class="sourceCode julia">limit(f(x)/g(x), x, a)</pre>
<div class="well well-sm">
$$\frac{16 a}{9}$$</div>
<h2>Questions</h2><h6>Question</h6><p>This function $f(x) = \sin(5x)/x$ is <em>indeterminate</em> at $x=0$. What type?</p><form name="WeaveQuestion" data-id="olyfyHdX" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_olyfyHdX" value="1"><div class="markdown">$0/0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_olyfyHdX" value="2"><div class="markdown">$\infty/\infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_olyfyHdX" value="3"><div class="markdown">$0^0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_olyfyHdX" value="4"><div class="markdown">$\infty - \infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_olyfyHdX" value="5"><div class="markdown">$0 \cdot \infty$
</div>
</label>
</div>
<div id="olyfyHdX_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_olyfyHdX']").on("change", function() {
correct = this.value == 1;
if(correct) {
$("#olyfyHdX_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#olyfyHdX_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>This function $f(x) = \sin(x)^{\sin(x)}$ is <em>indeterminate</em> at $x=0$. What type?</p><form name="WeaveQuestion" data-id="zPW5dLYz" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_zPW5dLYz" value="1"><div class="markdown">$0/0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_zPW5dLYz" value="2"><div class="markdown">$\infty/\infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_zPW5dLYz" value="3"><div class="markdown">$0^0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_zPW5dLYz" value="4"><div class="markdown">$\infty - \infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_zPW5dLYz" value="5"><div class="markdown">$0 \cdot \infty$
</div>
</label>
</div>
<div id="zPW5dLYz_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_zPW5dLYz']").on("change", function() {
correct = this.value == 3;
if(correct) {
$("#zPW5dLYz_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#zPW5dLYz_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>This function $f(x) = (x-2)/(x^2 - 4)$ is <em>indeterminate</em> at $x=2$. What type?</p><form name="WeaveQuestion" data-id="KwB8YIzW" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_KwB8YIzW" value="1"><div class="markdown">$0/0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_KwB8YIzW" value="2"><div class="markdown">$\infty/\infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_KwB8YIzW" value="3"><div class="markdown">$0^0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_KwB8YIzW" value="4"><div class="markdown">$\infty - \infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_KwB8YIzW" value="5"><div class="markdown">$0 \cdot \infty$
</div>
</label>
</div>
<div id="KwB8YIzW_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_KwB8YIzW']").on("change", function() {
correct = this.value == 1;
if(correct) {
$("#KwB8YIzW_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#KwB8YIzW_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>This function $f(x) = (g(x+h) - g(x-h)) / (2h)$ ($g$ is continuous) is <em>indeterminate</em> at $h=0$. What type?</p><form name="WeaveQuestion" data-id="tj7x7wpr" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_tj7x7wpr" value="1"><div class="markdown">$0/0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_tj7x7wpr" value="2"><div class="markdown">$\infty/\infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_tj7x7wpr" value="3"><div class="markdown">$0^0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_tj7x7wpr" value="4"><div class="markdown">$\infty - \infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_tj7x7wpr" value="5"><div class="markdown">$0 \cdot \infty$
</div>
</label>
</div>
<div id="tj7x7wpr_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_tj7x7wpr']").on("change", function() {
correct = this.value == 1;
if(correct) {
$("#tj7x7wpr_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#tj7x7wpr_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>This function $f(x) = x \log(x)$ is <em>indeterminate</em> at $x=0$. What type?</p><form name="WeaveQuestion" data-id="BzkOYalA" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_BzkOYalA" value="1"><div class="markdown">$0/0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_BzkOYalA" value="2"><div class="markdown">$\infty/\infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_BzkOYalA" value="3"><div class="markdown">$0^0$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_BzkOYalA" value="4"><div class="markdown">$\infty - \infty$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_BzkOYalA" value="5"><div class="markdown">$0 \cdot \infty$
</div>
</label>
</div>
<div id="BzkOYalA_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_BzkOYalA']").on("change", function() {
correct = this.value == 5;
if(correct) {
$("#BzkOYalA_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#BzkOYalA_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>Does L'Hopital's rule apply to this limit:</p>$$~
\lim_{x \rightarrow \pi} \frac{\sin(\pi x)}{\pi x}.
~$$<form name="WeaveQuestion" data-id="kQSzRwlh" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_kQSzRwlh" value="1"><div class="markdown"><p>Yes. It is of the form $0/0$</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_kQSzRwlh" value="2"><div class="markdown"><p>No. It is not indeterminate</p>
</div>
</label>
</div>
<div id="kQSzRwlh_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_kQSzRwlh']").on("change", function() {
correct = this.value == 2;
if(correct) {
$("#kQSzRwlh_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#kQSzRwlh_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>Use L'Hopital's rule to find the limit</p>$$~
L = \lim_{x \rightarrow 0} \frac{4x - \sin(x)}{x}.
~$$<p>What is $L$?</p><form name='WeaveQuestion' data-id='wTzFKGlV' data-controltype='numeric'>
<div class='form-group '>
<div class='controls'>
<div class="input-group">
<input id="wTzFKGlV" type="number" class="form-control">
</div>
<div id='wTzFKGlV_message'></div>
</div>
</div>
</form>
<script text='text/javascript'>
$('#wTzFKGlV').on('change', function() {
correct = Math.abs(this.value - 3.0) <= 0.001;
if(correct) {
$('#wTzFKGlV_message').html('<div class="alert alert-success"><span class="glyphicon glyphicon-thumbs-up"> Correct</span></div>');
} else {
$('#wTzFKGlV_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
}
});
</script>
<h6>Question</h6><p>Use L'Hopital's rule to find the limit</p>$$~
L = \lim_{x \rightarrow 0} \frac{\sqrt{1+x} - 1}{x}.
~$$<p>What is $L$?</p><form name='WeaveQuestion' data-id='KLEDVdO7' data-controltype='numeric'>
<div class='form-group '>
<div class='controls'>
<div class="input-group">
<input id="KLEDVdO7" type="number" class="form-control">
</div>
<div id='KLEDVdO7_message'></div>
</div>
</div>
</form>
<script text='text/javascript'>
$('#KLEDVdO7').on('change', function() {
correct = Math.abs(this.value - 0.5) <= 0.001;
if(correct) {
$('#KLEDVdO7_message').html('<div class="alert alert-success"><span class="glyphicon glyphicon-thumbs-up"> Correct</span></div>');
} else {
$('#KLEDVdO7_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
}
});
</script>
<h6>Question</h6><p>Use L'Hopital's rule <em>two</em> or more times to find the limit</p>$$~
L = \lim_{x \rightarrow 0} \frac{x - \sin(x)}{x^3}.
~$$<p>What is $L$?</p><form name='WeaveQuestion' data-id='68ZeLskk' data-controltype='numeric'>
<div class='form-group '>
<div class='controls'>
<div class="input-group">
<input id="68ZeLskk" type="number" class="form-control">
</div>
<div id='68ZeLskk_message'></div>
</div>
</div>
</form>
<script text='text/javascript'>
$('#68ZeLskk').on('change', function() {
correct = Math.abs(this.value - 0.16666666666666666) <= 0.001;
if(correct) {
$('#68ZeLskk_message').html('<div class="alert alert-success"><span class="glyphicon glyphicon-thumbs-up"> Correct</span></div>');
} else {
$('#68ZeLskk_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
}
});
</script>
<h6>Question</h6><p>Use L'Hopital's rule <em>two</em> or more times to find the limit</p>$$~
L = \lim_{x \rightarrow 0} \frac{1 - x^2/2 - \cos(x)}{x^3}.
~$$<p>What is $L$?</p><form name='WeaveQuestion' data-id='1WvWP4wT' data-controltype='numeric'>
<div class='form-group '>
<div class='controls'>
<div class="input-group">
<input id="1WvWP4wT" type="number" class="form-control">
</div>
<div id='1WvWP4wT_message'></div>
</div>
</div>
</form>
<script text='text/javascript'>
$('#1WvWP4wT').on('change', function() {
correct = Math.abs(this.value - 0.0) <= 0.001;
if(correct) {
$('#1WvWP4wT_message').html('<div class="alert alert-success"><span class="glyphicon glyphicon-thumbs-up"> Correct</span></div>');
} else {
$('#1WvWP4wT_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
}
});
</script>
<h6>Question</h6><p>By using a common denominator to rewrite this expression, use L'Hopital's rule to find the limit</p>$$~
L = \lim_{x \rightarrow 0} \frac{1}{x} - \frac{1}{\sin(x)}.
~$$<p>What is $L$?</p><form name='WeaveQuestion' data-id='jVulj5nc' data-controltype='numeric'>
<div class='form-group '>
<div class='controls'>
<div class="input-group">
<input id="jVulj5nc" type="number" class="form-control">
</div>
<div id='jVulj5nc_message'></div>
</div>
</div>
</form>
<script text='text/javascript'>
$('#jVulj5nc').on('change', function() {
correct = Math.abs(this.value - 0.0) <= 0.001;
if(correct) {
$('#jVulj5nc_message').html('<div class="alert alert-success"><span class="glyphicon glyphicon-thumbs-up"> Correct</span></div>');
} else {
$('#jVulj5nc_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
}
});
</script>
</div>
</div>
</body>
</html>