Problem:
Score : 600 points

Problem Statement

Kenkoooo found a simple connected graph.
The vertices are numbered 1 through n.
The i-th edge connects Vertex u_i and v_i, and has a fixed integer s_i.

Kenkoooo is trying to write a positive integer in each vertex so that the following condition is satisfied:

For every edge i, the sum of the positive integers written in Vertex u_i and v_i is equal to s_i.

Find the number of such ways to write positive integers in the vertices.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq u_i < v_i \leq n

2 \leq s_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

The graph is connected.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m
u_1 v_1 s_1
:
u_m v_m s_m

Output

Print the number of ways to write positive integers in the vertices so that the condition is satisfied.

Sample Input 1

3 3
1 2 3
2 3 5
1 3 4

Sample Output 1

1

The condition will be satisfied if we write 1,2 and 3 in vertices 1,2 and 3, respectively.
There is no other way to satisfy the condition, so the answer is 1.

Sample Input 2

4 3
1 2 6
2 3 7
3 4 5

Sample Output 2

3

Let a,b,c and d be the numbers to write in vertices 1,2,3 and 4, respectively.
There are three quadruples (a,b,c,d) that satisfy the condition:

(a,b,c,d)=(1,5,2,3)

(a,b,c,d)=(2,4,3,2)

(a,b,c,d)=(3,3,4,1)

Sample Input 3

8 7
1 2 1000000000
2 3 2
3 4 1000000000
4 5 2
5 6 1000000000
6 7 2
7 8 1000000000

Sample Output 3

0

Solution in PyPy3 (2.4.0) (judge verdict: Runtime Error)

def is_bipartite(G):
    v_min = [0, float('inf')]
    value = [-1]*n
    color = [-1]*n
    que = deque([0])
    color[0] = 0
    value[0] = 0
    while que:
        v = que.pop()
        val = value[v]
        c = color[v]
        for nv, s in G[v]:
            if color[nv]<0:
                color[nv] = 1-c
                value[nv] = s-val
                v_min[1-c] = min(v_min[1-c], s-val)
                que.append(nv)
            elif color[nv] == c:
                return (False, v_min, value[nv], s-val, nv)
    return (True, v_min, 0, 0, 0)
def C(v, c):
    value = [-1]*n
    que = deque([v])
    value[v] = c
                                                                                
    while que:
        v = que.pop()
        val = value[v]
        for nv, s in G[v]:
            if value[nv]<0:
                value[nv] = s-val
                if s-val<=0:
                    return False
                que.append(nv)
            elif value[nv] != s-val:
                return False
    return True

n, m = map(int, input().split())
G = [[] for _ in range(n)]
for _ in range(m):
    u, v, s = map(int, input().split())
    G[u-1].append((v-1, s))
    G[v-1].append((u-1, s))
color = [-1]*n
from collections import deque
flag, v_min, a, b, v = is_bipartite(G)
if flag:
    a, b = v_min
    if a+b<=1:
        print(0)
    else:
        print(a+b-1)
else:
    if (a+b)%2 != 0 or (a+b)<=0:
        print(0)
    else:
        c = (a+b)//2
        if C(v, c):
            print(agkbnf)
            print(1)
        else:
            print(0)