Score : $500$ points

Problem Statement

There are $N$ towns numbered $1, \dots, N$, and $M$ roads numbered $1, \dots, M$. Every road is directed; road $i$ $(1 \leq i \leq M)$ leads you from Town $A_i$ to Town $B_i$. The length of road $i$ is $C_i$.

You are given a sequence $E = (E_1, \dots, E_K)$ of length $K$ consisting of integers between $1$ and $M$. A way of traveling from town $1$ to town $N$ using roads is called a good path if:

• the sequence of the roads' numbers arranged in the order used in the path is a subsequence of $E$.

Note that a subsequence of a sequence is a sequence obtained by removing $0$ or more elements from the original sequence and concatenating the remaining elements without changing the order.

Find the minimum sum of the lengths of the roads used in a good path. If there is no good path, report that fact.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq M, K \leq 2 \times 10^5$
• $1 \leq A_i, B_i \leq N, A_i \neq B_i \, (1 \leq i \leq M)$
• $1 \leq C_i \leq 10^9 \, (1 \leq i \leq M)$
• $1 \leq E_i \leq M \, (1 \leq i \leq K)$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $K$

$A_1$ $B_1$ $C_1$

$\vdots$

$A_M$ $B_M$ $C_M$

$E_1$ $\ldots$ $E_K$

Output

Find the minimum sum of the lengths of the roads used in a good path. If there is no good path, print -1.

3 4 4
1 2 2
2 3 2
1 3 3
1 3 5
4 2 1 2

4

There are two possible good paths as follows:

• Using road $4$. In this case, the sum of the length of the used road is $5$.
• Using road $1$ and $2$ in this order. In this case, the sum of the lengths of the used roads is $2 + 2 = 4$.

Therefore, the desired minimum value is $4$.

3 2 3
1 2 1
2 3 1
2 1 1

-1

There is no good path.

4 4 5
3 2 2
1 3 5
2 4 7
3 4 10
2 4 1 4 3

14