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Contest: Task: Related: TaskC TaskE

Score : $700$ points

Problem Statement

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.

Sample Input 1

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

Sample Output 1

Yes

For example, consider a graph with edges $(0,1),(1,2),(1,4),(2,3),(2,4)$. This graph matches the clues.

Sample Input 2

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

Sample Output 2

No

Sample Input 3

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

Sample Output 3

No