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Problem #454

Automatic typewriter
 Public ★(x5) 01/08/18 by liuguangxi 10xp Probability 100.0%

An automatic typewriter can generate a random digit string by generating a sequence of random digits and concatenating them together. Each digit is chosen independently from 0 to 9 with probability 1/55, 2/55, 3/55, 4/55, 5/55, 6/55, 7/55, 8/55, 9/55 and 10/55, respectively. The generation is stopped as soon as a specific pattern occurs in the random string.

$D(n)$ is a string consisting of first $n$ digits of $\pi$. For example, $D(3) = 314$, $D(10) = 3141592653$.

Let $E(n)$ be the expected length of the generated random string for the string pattern $D(n)$. For example, $E(3) = 4159.375$, $E(10) = 104702034619.87625...$. Usually $E(n)$ is a decimal number.

Let $S(n)$ be $E(n)$ rounded to the nearest integer. You are given $S(3) = 4159$, $S(10) = 104702034620$ and $S(50) = 1812330404172820790558529569081392090013172561045583$.

Find S(10000).

As the answer is a very big number, use the following condensed representation:
(First 10 digits)[(number of remaining digits)](Last 10 digits)
For instance, the representation of 2127 is: 1701411834[19]5884105728.

Answer format: [condensed representation of S(10000)]

Example: 1812330404[32]2561045583 for $S(50)$.

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