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 C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

template<typename T>
void out(T x) { cout << x << endl; exit(0); }
#define watch(x) cout << (#x) << " is " << (x) << endl





using ll = long long;

const ll mod = 1e9+7;
const int maxn = 1e6 + 5;
const ll inf = 1e10;


int n, m;
vector<pair<ll,int>> g[maxn]; //s,to


bool test(ll x) {
    assert(x>0);
    vector<ll> a(n);
    a[0]=x;
    queue<int> qq;
    qq.push(0);
    while (qq.size()) {
	int at = qq.front(); qq.pop();
	for (auto ed: g[at]) {
	    ll wei = ed.first;
	    int to = ed.second;
	    if (a[to]>0) {
		if (a[to]+a[at]!=wei) return false;
	    } else {
		a[to] = wei-a[at];
		if (a[to]<=0) return false;
		qq.push(to);
	    }
	}
    }
    
    for (int i=0; i<n; i++) {
	if (a[i]==0) return false;
    }
    for (int i=0; i<n; i++) {
	for (auto ed: g[i]) {
	    int wei = ed.first;
	    int j = ed.second;
	    if (wei != a[i]+a[j]) return false;
	}
    }

    return true;
}

int main() {
    ios_base::sync_with_stdio(false); cin.tie(0);  cout.tie(0);

    cin>>n>>m;
    for (int i=0; i<m; i++) {
	int u,v,w; cin>>u>>v>>w;
	--u; --v;
	g[u].push_back({w,v});
	g[v].push_back({w,u});
    }

    ll L = -1, R = -1;

    // find upper-bound
    {
	ll lo = 0;
	ll hi = inf;


	while (hi-lo>1) {
	    ll mid = (lo+hi)/2;
	    if (test(mid)) {
		lo = mid;
	    } else {
		hi = mid;
	    }
	}
	if (lo==0) out(0);
	R = lo;
    }


    // find lower-bound
    {
	ll lo = 0;
	ll hi = R;


	while (hi-lo>1) {
	    ll mid = (lo+hi)/2;
	    if (test(mid)) {
		hi = mid;
	    } else {
		lo = mid;
	    }
	}

	L = hi;
    }

    ll res = R-L+1;
    cout<<res<<endl;    
    return 0;
}