Just Shuffle the Input

A permutation $p$ of size $n$ is a sequence of $n$ pairwise distinct numbers from $1$ to $n$. We write $p(i)$ for the $i$-th element. $p^k(i)$ denotes $p$ applied $k$ times, that is, $p(p(\ldots p(i)\ldots))$. A permutation is cyclic if the smallest positive integer $k$ with $p^k(1)=1$ equals $n$.

You are given a string $s$ of length $n$, a string $t$ of length $m$, and a cyclic permutation $p$ of size $m$. You want $t$ to become a substring of $s$. To do this, you may apply the shuffle operation zero or more times. The shuffle operation replaces $t$ with $t'$ such that the $i$-th letter of $t$ equals the $p(i)$-th letter of $t'$ for every $i$ from $1$ to $m$.

A string $a$ is a substring of $s$ if there exists an $l$ with $1 \le l \le |s|-|a|+1$ such that $s_{l+i-1}=a_i$ for every $i$ from $1$ to $|a|$.

Determine whether you can obtain a substring of $s$ from $t$. If you can, find the minimum number of shuffle operations needed.

The first line contains two integers $n$ and $m$ ($1 \le m \le n \le 200000$). The second line contains $m$ integers $p(1), \ldots, p(m)$, the permutation you can apply. The next two lines contain the string $s$ of length $n$ and the string $t$ of length $m$. Both strings consist of lowercase English letters. The given permutation is guaranteed to be cyclic.

If you cannot obtain a substring of $s$ from $t$, print $-1$. Otherwise, print the minimum number of shuffle operations needed.