Problem:
Score : 1300 points

Problem Statement

We have a graph with N vertices, numbered 0 through N-1. Edges are yet to be added.

We will process Q queries to add edges.
In the i-th (1≦i≦Q) query, three integers A_i, B_i and C_i will be given, and we will add infinitely many edges to the graph as follows:

The two vertices numbered A_i and B_i will be connected by an edge with a weight of C_i.

The two vertices numbered B_i and A_i+1 will be connected by an edge with a weight of C_i+1.

The two vertices numbered A_i+1 and B_i+1 will be connected by an edge with a weight of C_i+2.

The two vertices numbered B_i+1 and A_i+2 will be connected by an edge with a weight of C_i+3.

The two vertices numbered A_i+2 and B_i+2 will be connected by an edge with a weight of C_i+4.

The two vertices numbered B_i+2 and A_i+3 will be connected by an edge with a weight of C_i+5.

The two vertices numbered A_i+3 and B_i+3 will be connected by an edge with a weight of C_i+6.

...

Here, consider the indices of the vertices modulo N.
For example, the vertice numbered N is the one numbered 0, and the vertice numbered 2N-1 is the one numbered N-1.

The figure below shows the first seven edges added when N=16, A_i=7, B_i=14, C_i=1:

After processing all the queries, find the total weight of the edges contained in a minimum spanning tree of the graph.

Constraints

2≦N≦200,000

1≦Q≦200,000

0≦A_i,B_i≦N-1

1≦C_i≦10^9

Input

The input is given from Standard Input in the following format:

N Q
A_1 B_1 C_1
A_2 B_2 C_2
:
A_Q B_Q C_Q

Output

Print the total weight of the edges contained in a minimum spanning tree of the graph.

Sample Input 1

7 1
5 2 1

Sample Output 1

21

The figure below shows the minimum spanning tree of the graph:

Note that there can be multiple edges connecting the same pair of vertices.

Sample Input 2

2 1
0 0 1000000000

Sample Output 2

1000000001

Also note that there can be self-loops.

Sample Input 3

5 3
0 1 10
0 2 10
0 4 10

Sample Output 3

42

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <vector>
#include <algorithm>
#include <queue>
#include <utility>
using namespace std;
typedef long long ll;

int par[200010],sz[200010];
void init(int n){
	for(int i=0;i<n;i++){
		par[i] = i; sz[i] = 1;
	}
}

ll find(ll x){
	if(par[x]==x) return x;
	return par[x] = find(par[x]);
}

void unite(ll x, ll y){
	x = find(x); y = find(y);
	if(x==y) return;
	if(sz[x]<sz[y]) swap(x,y);
	par[y] = x; sz[x] += sz[y];
}

bool same(ll x, ll y){
	return find(x)==find(y);
}
typedef pair<ll,pair<ll,ll>> P;
priority_queue<P,vector<P>,greater<P>> que;
int main(){
	int i,n,q;
	cin >> n >> q;
	init(n);
	for(i=0;i<q;i++){
		ll a,b,c; cin >> a >> b >> c;
		que.push({c,{a,b}});
		que.push({c + 1,{(a + 1)%n,b}});
	}
	ll res = 0;
	while(que.size()){
		P p = que.top();
		que.pop();
		ll u = p.second.first,v = p.second.second;
		if(same(u,v)) continue;
		unite(u,v); res += p.first;
		p.first += 2; (p.second.first += 1) %= n; (p.second.second += 1) %= n;
		que.push(p);
	}
	cout << res << endl;
}