A non-decreasing sequence of integers $a_n$ can be generated from any positive
real value $\theta$ by the following procedure:
$$\begin{align}
\begin{split}
b_1 &= \theta \\\
b_n &= \left\lfloor b_{n-1} \right\rfloor
\left(b_{n-1} - \left\lfloor b_{n-1} \right\rfloor + 1\right)~~~
\forall ~ n \geq 2 \\\
a_n &= \left\lfloor b_{n} \right\rfloor
\end{split}
\end{align}$$
Where $\left\lfloor \cdot \right\rfloor$ is the floor function.
For example, $\theta=2.956938891377988...$ generates the Fibonacci sequence:
$2, 3, 5, 8, 13, 21, 34, 55, 89, ...$
The concatenation of a sequence of positive integers $a_n$ is a real value
denoted $\tau$ constructed by concatenating the elements of the sequence after
the decimal point, starting at $a_1$: $a_1.a_2a_3a_4...$
For example, the Fibonacci sequence constructed from
$\theta=2.956938891377988...$ yields the concatenation
$\tau=2.3581321345589...$ Clearly, $\tau \neq \theta$ for this value of
$\theta$.
Find the only value of $\theta$ for which the generated sequence starts at
$a_1=2$ and the concatenation of the generated sequence equals the original
value: $\tau = \theta$. Give your answer rounded to $24$ places after the
decimal point.