For every positive integer $n$ the Fibonacci sequence modulo $n$ is periodic. The period depends on the value of $n$. This period is called the Pisano period for $n$, often shortened to $\pi(n)$.

Define $M(p)$ as the largest integer $n$ such that $\pi(n) = p$, and define $M(p) = 1$ if there is no such $n$.
For example, there are three values of $n$ for which $\pi(n)$ equals $18$: $19, 38, 76$. Therefore $M(18) = 76$.

Let the sum function $S(n)$ be: $$S(n)=\sum_{p = 1}^{n} M(p).$$ You are given: $S(10)=26$.

Find $S(11^N) \bmod 1\,234\,567\,891$.