Contest Duration: - (local time) (100 minutes) Back to Home
D - joisino's travel /

Time Limit: 2 sec / Memory Limit: 256 MB

### 問題文

Atcoder国には N 個の町があり、M 本の双方向に移動可能な道で結ばれています。

また、 i 本目の道は町 A_i と町 B_i の間を距離 C_i で結んでいます。

joisinoお姉ちゃんは、この国の R 個の町 r_1,r_2..r_R を訪れることとなりました。

### 制約

• 2≦N≦200
• 1≦M≦N×(N-1)/2
• 2≦R≦min(8,N) ( min(8,N)8N のうち小さい方)
• r_i≠r_j (i≠j)
• 1≦A_i,B_i≦N , A_i≠B_i
• (A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)
• 1≦C_i≦100000
• すべての町の間は道のみで移動することができる。
• 入力は全て整数である。

### 入力

N M R
r_1 ... r_R
A_1 B_1 C_1
:
A_M B_M C_M


### 入力例 1

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


### 出力例 1

2


### 入力例 2

3 3 2
1 3
2 3 2
1 3 6
1 2 2


### 出力例 2

4


1 を訪れたあとに町 3 を訪れても、町 3 を訪れたあとに町 1 を訪れても、町 1 と町 3 の間の最短距離は 4 であるため、どの順番を選んだとしても答えは 4 となります。

### 入力例 3

4 6 3
2 3 4
1 2 4
2 3 3
4 3 1
1 4 1
4 2 2
3 1 6


### 出力例 3

3


Score : 400 points

### Problem Statement

There are N towns in the State of Atcoder, connected by M bidirectional roads.

The i-th road connects Town A_i and B_i and has a length of C_i.

Joisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).

She will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.

If she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?

### Constraints

• 2≤N≤200
• 1≤M≤N×(N-1)/2
• 2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)
• r_i≠r_j (i≠j)
• 1≤A_i,B_i≤N, A_i≠B_i
• (A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)
• 1≤C_i≤100000
• Every town can be reached from every town by road.
• All input values are integers.

### Input

Input is given from Standard Input in the following format:

N M R
r_1 ... r_R
A_1 B_1 C_1
:
A_M B_M C_M


### Output

Print the distance traveled by road if Joisino visits the towns in the order that minimizes it.

### Sample Input 1

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


### Sample Output 1

2


For example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.

### Sample Input 2

3 3 2
1 3
2 3 2
1 3 6
1 2 2


### Sample Output 2

4


The shortest distance between Towns 1 and 3 is 4. Thus, whether she visits Town 1 or 3 first, the distance traveled will be 4.

### Sample Input 3

4 6 3
2 3 4
1 2 4
2 3 3
4 3 1
1 4 1
4 2 2
3 1 6


### Sample Output 3

3