Derangement Rotations

A Derangement is a permutation $p$ of $1, 2, \dots , n$ where $p\_i \ne i$ for all $i$ from $1$ to $n$.

A rotation of a sequence $a\_1, a\_2, \dots , a\_n$ with offset $k$ ($1 \le k \le n$) is equal to the sequence $a\_k, a\_{k+1}, \dots, a\_n, a\_1, a\_2, \dots, a\_{k-1}$. A sequence of length $n$ has at most $n$ distinct rotations.

Given a derangement $D$, let $f(D)$ denote the number of distinct rotations of $D$ that are also derangements. For example, $f(\[2, 1]) = 1$, $f(\[3, 1, 2]) = 2$.

Given $n$ and a prime number $p$, count the number of derangements $D$ of $1, 2, \dots , n$ such that $f(D) = n - 2$, modulo $p$.

The single line of input contains two integers $n$ ($3 \le n \le 10^6$) and $p$ ($10^8 \le p \le 10^9 + 7$), where $n$ is a permutation size, and $p$ is a prime number.

Output a single integer, which is the number of derangements $D$ of size $n$ with $f(D) = n - 2$, modulo $p$.