Consider the infinite repeating sequence of digits:
1234321234321234321...
Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the digits in the $n$-th value is $n$.
The sequence goes as follows:
1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, ...
Let $v_n$ be the $n$-th value in this sequence. For example, $v_2=2$, $v_5=32$ and $v_{11}=32123$.
Let $S(n)$ be $v_1+v_2+\cdots+v_n$. For example, $S(11)=36120$, and $S(1000)\bmod 123454321=18232686$.
Find $S(10^{14})\bmod 123454321$.
Solution
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