Problem Statement

There are N kinds of magical gems, numbered $1, 2, \ldots, N,$ distributed in the AtCoder Kingdom.
Takahashi is trying to make an ornament by arranging gems in a row.
For some pairs of gems, we can put the two gems next to each other; for other pairs, we cannot. We have M pairs for which the two gems can be adjacent: (Gem $A_1, Gem B_1), (Gem A_2, Gem B_2), \ldots, (Gem A_M, Gem B_M)$. For the other pairs, the two gems cannot be adjacent. (Order does not matter in these pairs.)
Determine whether it is possible to form a sequence of gems that has one or more gems of each of the kinds $C_1, C_2, \dots, C_K$. If the answer is yes, find the minimum number of stones needed to form such a sequence.

Input

Input is given from Standard Input in the following format:
N M
A_1 B_1
A_2 B_2
$\hspace{7mm}\vdots$
A_M B_M
K
$C_1 C_2 \cdots C_K$

Output

Print the minimum number of stones needed to form a sequence of gems that has one or more gems of each of the kinds $C_1, C_2, \dots, C_K$.
If such a sequence cannot be formed, print -1 instead.

Constraints

• All values in input are integers.
• 1 ≤ N ≤ 10^5
• 0 ≤ M ≤ 10^5
• 1 ≤ A_i < B_i ≤ N
• If i ≠ j, (A_i, B_i) ≠ (A_j, B_j).
• 1 ≤ K ≤ 17
• 1 ≤ $C_1 < C_2 < \dots < C_K$ ≤ N

Sample Input 1

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

Sample Output 1

5

For example, by arranging the gems in the order [1, 4, 2, 4, 3], we can form a sequence of length 5 with Gems 1, 2, 3.

Sample Input 2

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

Sample Output 2

-1

Sample Input 3

10 10
3 9
3 8
8 10
2 10
5 8
6 8
5 7
6 7
1 6
2 4
4
1 2 7 9

Sample Output 3

11

For example, by arranging the gems in the order [1, 6, 7, 5, 8, 3, 9, 3, 8, 10, 2], we can form a sequence of length 11 with Gems 1, 2, 7, 9.

见题面 Input 部分

见题面 Output 部分