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Contest: Task: Related: TaskF TaskH

Score : $600$ points

Problem Statement

We have a simple connected undirected graph with $N$ vertices and $M$ edges.
The vertices are numbered as Vertex $1$, Vertex $2$, $\dots$, Vertex $N$.
The edges are numbered as Edge $1$, Edge $2$, $\dots$, Edge $M$. Edge $i$ connects Vertex $a_i$ and Vertex $b_i$ bidirectionally. There is no edge that directly connects Vertex $1$ and Vertex $N$.
Each vertex is either empty or occupied by a wall. Initially, every vertex is empty.

Aoki is going to travel from Vertex $1$ to Vertex $N$ along the edges on the graph. However, Aoki is not allowed to move to a vertex occupied by a wall.

Takahashi has decided to choose some of the vertices to build walls on, so that Aoki cannot travel to Vertex $N$ no matter which route he takes.
Building a wall on Vertex $i$ costs Takahashi $c_i$ yen (the currency of Japan). He cannot build a wall on Vertex $1$ and Vertex $N$.

How many yens is required for Takahashi to build walls so that the conditions is satisfied? Also, print the way of building walls to achieve the minimum cost.

Constraints

  • $3 \leq N \leq 100$
  • $N - 1 \leq M \leq \frac{N(N-1)}{2} - 1$
  • $1 \leq a_i \lt b_i \leq N$ $(1 \leq i \leq M)$
  • $(a_i, b_i) \neq (1, N)$
  • The given graph is simple and connected.
  • $1 \leq c_{i} \leq 10^9$ $(2 \leq i \leq N-1)$
  • $c_1 = c_N = 0$
  • All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$a_1$ $b_1$
$a_2$ $b_2$
$\vdots$
$a_M$ $b_M$
$c_1$ $c_2$ $\dots$ $c_N$

Output

Print in the following format. Here, $C,k$, and $p_i$ are defined as follows.

  • $C$ is the cost that Takahashi will pay
  • $k$ is the number of vertices for Takahashi to build walls on
  • $(p_1,p_2,\dots,p_k)$ is a sequence of vertices on which Takahashi will build walls
$C$
$k$
$p_1$ $p_2$ $\dots$ $p_k$

If there are multiple ways to build walls to satisfy the conditions with the minimum cost, print any of them.

Sample Input 1

5 5
1 2
2 3
3 5
2 4
4 5
0 8 3 4 0

Sample Output 1

7
2
3 4

If Takahashi builds walls on Vertex $3$ and Vertex $4$, paying $3 + 4 = 7$ yen, Aoki is unable to travel from Vertex $1$ to Vertex $5$.
There is no way to satisfy the condition with less cost, so $7$ yen is the answer.

Sample Input 2

3 2
1 2
2 3
0 1 0

Sample Output 2

1
1
2

Sample Input 3

5 9
1 2
1 3
1 4
2 3
2 4
2 5
3 4
3 5
4 5
0 1000000000 1000000000 1000000000 0

Sample Output 3

3000000000
3
2 3 4