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854. Pisano Periods 2

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 product function $P(n)$ be: $$P(n)=\prod_{p = 1}^{n}M(p).$$ You are given: $P(10)=264$.

Find $P(1\,000\,000)\bmod 1\,234\,567\,891$.