#include <bits/stdc++.h>
 
#ifdef NON_SUBMIT
#define TEST(n) (n)
#define tout cerr
#else
#define TEST(n) ((void)0)
#define tout cin
#endif
 
using namespace std;
 
vector<pair<int,int>> E;
int color[100000];
bool chk[100000];
 
int find_(int a)
{
	return color[a]=a==color[a] ? a:find_(color[a]);
}
 
void union_(int a, int b)
{
	a=find_(a); b=find_(b);
	color[a]=color[b]=max(a,b);
}
 
int main()
{
	ios::sync_with_stdio(false);
	cin.tie(NULL); cout.tie(NULL);
	TEST(freopen("input.txt","r",stdin));
	TEST(freopen("output.txt","w",stdout));
	TEST(freopen("debug.txt","w",stderr));
	int N, Q, c0=0, c1=0;
	long long M;
	cin>>N>>M>>Q;
	for(int i=0;i<N;i++) color[i]=i;
	while(Q--) {
		int a, b, c;
		cin>>a>>b>>c;
		if(c==0) {
			if(find_(a)!=find_(b)) {
				union_(a,b);
				M--;
			}
		}
		else E.emplace_back(a,b);
	}
	for(auto e: E) {
		int a, b;
		tie(a,b)=e;
		if(find_(a)==find_(b)) {
			cout<<"No\n";
			return 0;
		}
		chk[find_(a)]=chk[find_(b)]=true;
	}
	for(int i=0;i<N;i++) {
		c0+=find_(i)==i;
		c1+=chk[i];
	}
	cout<<(M>=0 && (c1==0 || c0>=3 && max(c1,3)<=M) && c0-1<=M && 1LL*c0*(c0-1)/2>=M ? "Yes\n":"No\n");
	return 0;
}
### Prompt

Generate a Cpp solution to the following problem:

Snuke's mother gave Snuke an undirected graph consisting of N vertices numbered 0 to N-1 and M edges. This graph was connected and contained no parallel edges or self-loops.

One day, Snuke broke this graph. Fortunately, he remembered Q clues about the graph. The i-th clue (0 \leq i \leq Q-1) is represented as integers A_i,B_i,C_i and means the following:

* If C_i=0: there was exactly one simple path (a path that never visits the same vertex twice) from Vertex A_i to B_i.
* If C_i=1: there were two or more simple paths from Vertex A_i to B_i.



Snuke is not sure if his memory is correct, and worried whether there is a graph that matches these Q clues. Determine if there exists a graph that matches Snuke's memory.

Constraints

* 2 \leq N \leq 10^5
* N-1 \leq M \leq N \times (N-1)/2
* 1 \leq Q \leq 10^5
* 0 \leq A_i,B_i \leq N-1
* A_i \neq B_i
* 0 \leq C_i \leq 1
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N M Q
A_0 B_0 C_0
A_1 B_1 C_1
\vdots
A_{Q-1} B_{Q-1} C_{Q-1}


Output

If there exists a graph that matches Snuke's memory, print `Yes`; otherwise, print `No`.

Examples

Input

5 5 3
0 1 0
1 2 1
2 3 0


Output

Yes


Input

4 4 3
0 1 0
1 2 1
2 3 0


Output

No


Input

10 9 9
7 6 0
4 5 1
9 7 0
2 9 0
2 3 0
4 1 0
8 0 0
9 1 0
3 0 0


Output

No

### Response

```cpp
#include <bits/stdc++.h>
 
#ifdef NON_SUBMIT
#define TEST(n) (n)
#define tout cerr
#else
#define TEST(n) ((void)0)
#define tout cin
#endif
 
using namespace std;
 
vector<pair<int,int>> E;
int color[100000];
bool chk[100000];
 
int find_(int a)
{
	return color[a]=a==color[a] ? a:find_(color[a]);
}
 
void union_(int a, int b)
{
	a=find_(a); b=find_(b);
	color[a]=color[b]=max(a,b);
}
 
int main()
{
	ios::sync_with_stdio(false);
	cin.tie(NULL); cout.tie(NULL);
	TEST(freopen("input.txt","r",stdin));
	TEST(freopen("output.txt","w",stdout));
	TEST(freopen("debug.txt","w",stderr));
	int N, Q, c0=0, c1=0;
	long long M;
	cin>>N>>M>>Q;
	for(int i=0;i<N;i++) color[i]=i;
	while(Q--) {
		int a, b, c;
		cin>>a>>b>>c;
		if(c==0) {
			if(find_(a)!=find_(b)) {
				union_(a,b);
				M--;
			}
		}
		else E.emplace_back(a,b);
	}
	for(auto e: E) {
		int a, b;
		tie(a,b)=e;
		if(find_(a)==find_(b)) {
			cout<<"No\n";
			return 0;
		}
		chk[find_(a)]=chk[find_(b)]=true;
	}
	for(int i=0;i<N;i++) {
		c0+=find_(i)==i;
		c1+=chk[i];
	}
	cout<<(M>=0 && (c1==0 || c0>=3 && max(c1,3)<=M) && c0-1<=M && 1LL*c0*(c0-1)/2>=M ? "Yes\n":"No\n");
	return 0;
}
```