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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
#define FOR(i,a,b) for(int i=(a);i<(b);++i)
#define REP(i,n) FOR(i,0,n)
#define ALL(v) begin(v),end(v)
#define fi first
#define se second
template<typename A, typename B> inline bool chmax(A &a, B b) { if (a<b) { a=b; return 1; } return 0; }
template<typename A, typename B> inline bool chmin(A &a, B b) { if (a>b) { a=b; return 1; } return 0; }
using ll = long long;
using pii = pair<int, int>;
constexpr ll INF = 1ll<<30;
constexpr ll longINF = 1ll<<60;
constexpr ll MOD = 1000000007;
constexpr bool debug = 0;
//---------------------------------//

struct UnionFind {
public:
	using size_type = std::size_t;
	
	UnionFind(size_type n) : n(n), size_(n, 1) {
		par.resize(n);
		std::iota(par.begin(), par.end(), 0);
	}
	
	size_type size(size_type x) { return size_[find(x)]; }
	
	size_type find(size_type x) {
		assert(x < n);
		while (par[x] != x) {
			par[x] = par[par[x]];
			x = par[x];
		}
		return x;
	}
	
	void unite(size_type x, size_type y) {
		x = find(x);
		y = find(y);
		if (x == y) return;
		if (size(x) > size(y)) std::swap(x, y);
		par[x] = y;
		size_[y] += size_[x];
	}
	
	bool issame(size_type x, size_type y) { return find(x) == find(y); }
	
private:
	size_type n;
	std::vector<size_type> size_, par;
};

int main() {
	int N, M;
	cin >> N >> M;
	vector<vector<pii>> g(N);
	vector<int> U(M), V(M), S(M);
	UnionFind uf(N);
	
	REP(i, M) {
		scanf("%d %d %d", &U[i], &V[i], &S[i]);
		--U[i]; --V[i];
		g[U[i]].emplace_back(V[i], i);
		g[V[i]].emplace_back(U[i], i);
		uf.unite(U[i], V[i]);
	}
	
	vector<pii> itv(N, pii(1, INF));
	REP(i, N) for (pii e : g[i]) chmin(itv[i].se, S[e.se] - 1);
	
	priority_queue<pii, vector<pii>, greater<pii>> pq;
	REP(i, N) pq.emplace(itv[i].se - itv[i].fi + 1, i);
	
	while (!pq.empty()) {
		pii cur = pq.top(); pq.pop();
		int u = cur.se;
		if (itv[u].se - itv[u].fi + 1 < cur.fi) continue;
		
		for (pii e : g[u]) {
			int v = e.fi;
			pii nex_itv(max(S[e.se] - itv[u].se, itv[v].fi), min(S[e.se] - itv[u].fi, itv[v].se));
			
			if (nex_itv.fi > nex_itv.se) {
				puts("0");
				return 0;
			}
			if (nex_itv.se - nex_itv.fi + 1 < itv[v].se - itv[v].fi + 1) {
				itv[v] = nex_itv;
				pq.emplace(nex_itv.se - nex_itv.fi + 1, v);
			}
		}
	}
	
	ll ans = 1;
	REP(i, N) if (uf.find(i) == i && uf.size(i) > 1) ans *= itv[i].se - itv[i].fi + 1;
	cout << ans << endl;
	return 0;
}