Pitch-Black Room

A wakes up inside a pitch-black room. He seems to have wandered into a dungeon made of $N$ rooms. You could not find out which room A is in, but luckily you got hold of the dungeon map. Point out the path A should take and lead him to a bright room.

Of the $N$ rooms, $M$ are pitch-black, and you know that rooms $D_1$, $D_2$, $...$, $D_M$ are the pitch-black ones. Also, every room has exactly $K$ one-way paths laid out in order, and the paths leaving room $i$ lead to rooms $v_{i,1}$, $v_{i,2}$, ..., $v_{i,K}$. You can make A take the $a_1$-th, $a_2$-th, ..., $a_l$-th paths in order from the room he is in. Once A reaches a bright room, he ignores all further instructions. You cannot know which room A is in, neither before nor after giving instructions, so the sequence of instructions you pass on must lead A to a bright room no matter which room he started in.

Among such instruction sequences, find the length of the shortest one.

The input is given from standard input in the following format.

$N$ $M$ $K$

$D_1$ $D_2$ $...$ $D_M$

$v_{1,1}$ $v_{1,2}$ $...$ $v_{1,K}$

$v_{2,1}$ $v_{2,2}$ $...$ $v_{2,K}$

$...$

$v_{N,1}$ $v_{N,2}$ $...$ $v_{N,K}$

Print the answer on one line.

• $2 \leq N \leq 100$
• $1 \leq M \leq min(16, N-1)$
• $1 \leq K \leq N$
• $1 \leq D_i \leq N$
• All $D_i$ are distinct
• $1 \leq v_{i, j} \leq N$
• Every dark room can reach at least one bright room