201444: [AtCoder]ARC144 E - GCD of Path Weights

Problem Statement

You are given a directed graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1, 2, \ldots, N$. The $i$-th edge is directed from Vertex $a_i$ to Vertex $b_i$, where $a_i < b_i$.

The beautifulness of a sequence of positive integers $W = (W_1, W_2, \ldots, W_N)$ is defined as the maximum positive integer $x$ that satisfies the following:

• For every path $(v_1, \ldots, v_k)$ ($v_1 = 1, v_k = N$) from Vertex $1$ to Vertex $N$ in $G$, $\sum_{i=1}^k W_{v_i}$ is a multiple of $x$.

You are given an integer sequence $A = (A_1, A_2, \ldots, A_N)$. Find the maximum beautifulness of a sequence of positive integers $W = (W_1, \ldots, W_N)$ such that $A_i \neq -1 \implies W_i = A_i$. If the maximum beautifulness does not exist, print -1.

Constraints

• $2\leq N\leq 3\times 10^5$
• $1\leq M\leq 3\times 10^5$
• $1\leq a_i < b_i \leq N$
• $(a_i,b_i)\neq (a_j,b_j)$ if $i\neq j$
• In the given graph $G$, there is a path from Vertex $1$ to Vertex $N$.
• $A_i = -1$ or $1\leq A_i\leq 10^{12}$

Input

Input is given from Standard Input in the following format:

$N$ $M$
$a_1$ $b_1$
$\vdots$
$a_M$ $b_M$
$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the maximum beautifulness of a sequence of positive integers $W$. If the maximum beautifulness does not exist, print -1.

Sample Input 1

4 4
1 2
1 3
2 4
3 4
-1 3 7 -1

Sample Output 1

4

There are two paths from Vertex $1$ to Vertex $N$: $(1,2,4)$ and $(1,3,4)$. For instance, $W = (5, 3, 7, 8)$ has a beautifulness of $4$. Indeed, both $W_1 + W_2 + W_4 = 16$ and $W_1 + W_3 + W_4 = 20$ are multiples of $4$.

Sample Input 2

4 5
1 2
1 3
2 4
3 4
1 4
-1 3 7 -1

Sample Output 2

1

There are three paths from Vertex $1$ to Vertex $N$: $(1,2,4)$, $(1,3,4)$, and $(1,4)$. For instance, $W = (5, 3, 7, 8)$ has a beautifulness of $1$.

Sample Input 3

4 4
1 2
1 3
2 4
3 4
3 -1 -1 7

Sample Output 3

-1

For instance, $W = (3, 10^{100}, 10^{100}, 7)$ has a beautifulness of $10^{100}+10$. Since you can increase the beautifulness of $W$ as much as you want, there is no maximum beautifulness.

Sample Input 4

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

Sample Output 4

9