Problem K
Deterministic Finite Automata - Minimum Word Length

You are given a deterministic finite automaton that accepts the language $\mathcal{L}$. You should output the length of the shortest word $w$ such that $w\in \mathcal{L}$ is a finite language. You may assume that $\mathcal{L}$ is non-empty.

Input

The input contains the description of a deterministic finite automaton.

The first line contains four positive integers $n$, $c$, $s$, and $f$, where $n$ is the number of states, $c$ is the size of the alphabet, $s$ is the initial state, and $f$ is the number of final states. The second line consists of a string $\mathrm{\Sigma }={\mathrm{\Sigma }}_1{\mathrm{\Sigma }}_2\dots {\mathrm{\Sigma }}_c$ of $c$ distinct symbols, each of which is a lowercase english character. The third line consists of $f$ distinct positive integers, the set of final states. Then $n$ lines follow, each with $c$ positive integers, describing the symbol table. The $j$-th integer on the $i$-th of those lines represents the state transitioned to from state $i$ upon reading ${\mathrm{\Sigma }}_j$.

Each state is an integer between $1$ and $n$. It is guaranteed that $1\le n\le 10000$, $1\le s\le n$, and $0\le f\le n$.

Output

Output the length of the shortest word in $\mathcal{L}$.

4 2 1 2
ab
2 4
2 3
3 2
3 3
1 2

1

4 4 1 2
acgt
1 2
2 2 2 2
3 3 3 3
4 4 3 4
3 3 3 4

0