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;
using ll = long long;

#define REP(i,n) for(long long i = 0; i < (n); i++)
#define FOR(i, m, n) for(long long i = (m);i < (n); ++i)
#define ALL(obj) (obj).begin(),(obj).end()

template<class T> using V = vector<T>;
template<class T, class U> using P = pair<T, U>;

const ll MOD = (ll)1e9 + 7;
const ll MOD2 = 998244353;
const ll LLINF = (ll)1e18;
const ll INTINF = (ll)1e9;
const long double PI = 3.1415926535897932384626433;

template <class T> void corner(bool flg, T hoge) {if (flg) {cout << hoge << endl; exit(0);}}
template <class T, class U>ostream &operator<<(ostream &o, const map<T, U>&obj) {o << "{"; for (auto &x : obj) o << " {" << x.first << " : " << x.second << "}" << ","; o << " }"; return o;}
template <class T>ostream &operator<<(ostream &o, const set<T>&obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr) o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
template <class T>ostream &operator<<(ostream &o, const vector<T>&obj) {o << "{"; for (int i = 0; i < (int)obj.size(); ++i)o << (i > 0 ? ", " : "") << obj[i]; o << "}"; return o;}
template <class T, class U>ostream &operator<<(ostream &o, const pair<T, U>&obj) {o << "{" << obj.first << ", " << obj.second << "}"; return o;}
template <template <class tmp>  class T, class U> ostream &operator<<(ostream &o, const T<U> &obj) {o << "{"; for (auto itr = obj.begin(); itr != obj.end(); ++itr)o << (itr != obj.begin() ? ", " : "") << *itr; o << "}"; return o;}
void print(void) {cout << endl;}
template <class Head> void print(Head&& head) {cout << head;print();}
template <class Head, class... Tail> void print(Head&& head, Tail&&... tail) {cout << head << " ";print(forward<Tail>(tail)...);}
template <class T> void chmax(T& a, const T b){a=max<T>(a,b);}
template <class T> void chmin(T& a, const T b){a=min<T>(a,b);}
void YN(bool flg) {cout << ((flg) ? "YES" : "NO") << endl;}
void Yn(bool flg) {cout << ((flg) ? "Yes" : "No") << endl;}
void yn(bool flg) {cout << ((flg) ? "yes" : "no") << endl;}

int main() {
	int N,M; cin >> N >> M;
	ll mini = 1, maxi = 0, x = 0;
	V<V<P<int,ll>>> edge(N);
	V<P<int,ll>> node(N,{0,0});
	for(int i = 0; i < M; ++i){
		int u,v; ll s;
		cin >> u >> v >> s;
		u--,v--;
		edge[u].push_back({v,s});
		edge[v].push_back({u,s});
		chmax(maxi,s-1);
	}

	node[0] = {1,0};
	queue<int> q;
	q.push(0);
	while(q.size()){
		int from = q.front();
		q.pop();
		for(auto e:edge[from]){
			int to = e.first;
			ll cost = e.second;
			int type = -node[from].first;
			ll val = cost-node[from].second;

			if(node[to].first==0){
				node[to] = {type,val};
				if(node[to].first==-1) chmin(maxi,node[to].second - 1);
				if(node[to].first== 1) chmax(mini,1 - node[to].second);
				q.push(to);
			}
			else{
				if(type == node[to].first){
					corner(val != node[to].second,0);
				}
				else{
					ll a = val, b = node[to].second;
					if(type > node[to].first) swap(a,b);
					corner(a-b<=0||(a-b)%2,0);
					x = (a-b)/2;
					while(q.size())q.pop();
					break;
				}
			}
		}
	}

	corner(x==0,max(maxi-mini+1LL,0LL));

	for(int i = 0; i < N; ++i) node[i] = {0,0};
	node[0] = {1,x};
	q.push(0);

	while(q.size()){
		int from = q.front();
		q.pop();
		for(auto e:edge[from]){
			int to = e.first;
			ll cost = e.second;
			ll val = cost-node[from].second;

			if(node[to].first==0){
				node[to] = {1,val};
				q.push(to);
			}
			else{
				corner(val != node[to].second, 0);
			}
		}
	}
	cout << 1 << endl;	
	return 0;
}