#include <iostream>
#include <vector>

using namespace std;

class UnionFind {
public:
    explicit UnionFind(int N) : root(N, -1), size(N, 1) {}
    int getRoot(int u){ return root[u] == -1 ? u : root[u] = getRoot(root[u]); }
    int getSize(int u){ return size[getRoot(u)]; }
    bool same(int a, int b){
        return getRoot(a) == getRoot(b);
    }
    bool merge(int a, int b){
        int u = getRoot(a);
        int v = getRoot(b);
        if(u == v) return false;
        root[u] = v;
        size[v] += size[u];
        return true;
    }
private:
    vector<int> root;
    vector<int> size;
};

bool solve(){
    long long N, M, Q; cin >> N >> M >> Q;
    UnionFind uf(N);
    vector<pair<int, int>> bi;
    long long sz = N;
    for(int i=0;i<Q;i++){
        int a, b, c; cin >> a >> b >> c;
        if(M == N-1 && c == 1) return false;
        if(c == 0){
            if(uf.merge(a, b)) --sz;
        } else {
            bi.emplace_back(a, b);
        }
    }
    if(M == N-1) return true;
    if(!bi.empty() && sz <= 2) return false;
    for(auto& p : bi){
        if(uf.same(p.first, p.second)) return false;
    }
    return N-sz + sz*(sz-1)/2 >= M;
}

int main(){
    cout << (solve() ? "Yes" : "No") << endl;
}
### Prompt

Create a solution in CPP for 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 <iostream>
#include <vector>

using namespace std;

class UnionFind {
public:
    explicit UnionFind(int N) : root(N, -1), size(N, 1) {}
    int getRoot(int u){ return root[u] == -1 ? u : root[u] = getRoot(root[u]); }
    int getSize(int u){ return size[getRoot(u)]; }
    bool same(int a, int b){
        return getRoot(a) == getRoot(b);
    }
    bool merge(int a, int b){
        int u = getRoot(a);
        int v = getRoot(b);
        if(u == v) return false;
        root[u] = v;
        size[v] += size[u];
        return true;
    }
private:
    vector<int> root;
    vector<int> size;
};

bool solve(){
    long long N, M, Q; cin >> N >> M >> Q;
    UnionFind uf(N);
    vector<pair<int, int>> bi;
    long long sz = N;
    for(int i=0;i<Q;i++){
        int a, b, c; cin >> a >> b >> c;
        if(M == N-1 && c == 1) return false;
        if(c == 0){
            if(uf.merge(a, b)) --sz;
        } else {
            bi.emplace_back(a, b);
        }
    }
    if(M == N-1) return true;
    if(!bi.empty() && sz <= 2) return false;
    for(auto& p : bi){
        if(uf.same(p.first, p.second)) return false;
    }
    return N-sz + sz*(sz-1)/2 >= M;
}

int main(){
    cout << (solve() ? "Yes" : "No") << endl;
}
```