#include<bits/stdc++.h>
#define MAXN 100005
#define INF 1000000000
#define MOD 1000000007
#define F first
#define S second
using namespace std;
typedef long long ll;
typedef pair<int,int> P;
int n,q,cnt;
ll m;
int p[MAXN],r[MAXN];
void init(int n)
{
    for(int i=0;i<n;i++)
    {
        p[i]=i;
        r[i]=0;
    }
    cnt=n;
}
int find(int x)
{
    if(p[x]==x) return x;
    else return p[x]=find(p[x]);
}
void unite(int x,int y)
{
    x=find(x);
    y=find(y);
    if(x==y) return;
    cnt--;
    if(r[x]<r[y]) p[x]=y;
    else
    {
        p[y]=x;
        if(r[x]==r[y]) r[x]++;
    }
}
bool same(int x,int y)
{
    return find(x)==find(y);
}
vector<P> con;
vector<P> ncon;
int main()
{
    scanf("%d%lld%d",&n,&m,&q);
    init(n);
    for(int i=0;i<q;i++)
    {
        int x,y,z;
        scanf("%d%d%d",&x,&y,&z);
        if(!z) con.push_back(P(x,y));
        else ncon.push_back(P(x,y));
    }
    for(auto p:con) unite(p.F,p.S);
    bool f=true;
    for(auto p:ncon)
    {
        if(same(p.F,p.S)) f=false;
    }
    if(!f) {puts("No"); return 0;}
    if(m==n-1)
    {
        if(!ncon.size()) puts("Yes"); else puts("No");
    }
    else
    {
        if(m<=n+1LL*cnt*(cnt-3)/2) puts("Yes"); else puts("No");
    }
    return 0;
}

### Prompt

Your challenge is to write 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>
#define MAXN 100005
#define INF 1000000000
#define MOD 1000000007
#define F first
#define S second
using namespace std;
typedef long long ll;
typedef pair<int,int> P;
int n,q,cnt;
ll m;
int p[MAXN],r[MAXN];
void init(int n)
{
    for(int i=0;i<n;i++)
    {
        p[i]=i;
        r[i]=0;
    }
    cnt=n;
}
int find(int x)
{
    if(p[x]==x) return x;
    else return p[x]=find(p[x]);
}
void unite(int x,int y)
{
    x=find(x);
    y=find(y);
    if(x==y) return;
    cnt--;
    if(r[x]<r[y]) p[x]=y;
    else
    {
        p[y]=x;
        if(r[x]==r[y]) r[x]++;
    }
}
bool same(int x,int y)
{
    return find(x)==find(y);
}
vector<P> con;
vector<P> ncon;
int main()
{
    scanf("%d%lld%d",&n,&m,&q);
    init(n);
    for(int i=0;i<q;i++)
    {
        int x,y,z;
        scanf("%d%d%d",&x,&y,&z);
        if(!z) con.push_back(P(x,y));
        else ncon.push_back(P(x,y));
    }
    for(auto p:con) unite(p.F,p.S);
    bool f=true;
    for(auto p:ncon)
    {
        if(same(p.F,p.S)) f=false;
    }
    if(!f) {puts("No"); return 0;}
    if(m==n-1)
    {
        if(!ncon.size()) puts("Yes"); else puts("No");
    }
    else
    {
        if(m<=n+1LL*cnt*(cnt-3)/2) puts("Yes"); else puts("No");
    }
    return 0;
}

```