Description

Input

The first line contains two integers $n$ and $m$ ($1\le n\le 20$, $0\le m\le n^2+n$).

Then following $m$ lines of input contains three integers $t$, $i$ and $j$ ($t \in \{0,1\}$, $1\le i\le j\le n$).

• If $t=0$, the condition is $x_i+x_j\le 1$.
• If $t=1$, the condition is $x_i+x_j\ge 1$.

It is guaranteed that all conditions are pairwise distinct.

Output

Output the probability that all the conditions are satisfied, modulo $M = 998~244~353$.

3 2
0 1 2
1 3 3

748683265

3 3
0 1 2
0 1 3
0 2 3

748683265

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

935854081

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

0

8 12
0 1 2
0 2 3
1 3 4
0 1 4
0 5 6
0 6 7
1 7 8
0 5 8
1 3 7
1 3 8
1 4 7
1 4 8

997687297

Note

In the first test case, the conditions are $x_1+x_2 \le 1$ and $x_3+x_3\ge 1$, and the probability that each condition is satisfied is $\frac 12$, so the probability that they are both satisfied is $\frac 12\cdot \frac 12=\frac 14$, modulo $998~244~353$ is equal to $748683265$.

In the second test case, the answer is $\frac 14$.

In the third test case, the answer is $\frac 1{16}$.

In the fourth test case, the sum of all variables must equal $2$, so the probability is $0$.