Score : $600$ points

Problem Statement

We have a graph $G$ with $N$ vertices numbered $1$ through $N$ and $0$ edges. You are given sequences $u=(u_1,u_2,\ldots,u_M),v=(v_1,v_2,\ldots,v_M)$ of length $M$.

You will perform the following operation $(N-1)$ times.

• Choose $i$ $(1 \leq i \leq M)$ uniformly at random. Add to $G$ an undirected edge connecting Vertices $u_i$ and $v_i$.

Note that the operation above will add a new edge connecting Vertices $u_i$ and $v_i$ even if $G$ already has one or more edges between them. In other words, the resulting $G$ may contain multiple edges.

For each $K=1,2,\ldots,N-1$, find the probability, modulo $998244353$, that $G$ is a forest after the $K$-th operation.

Constraints

• $2 \leq N \leq 14$
• $N-1 \leq M \leq 500$
• $1 \leq u_i,v_i \leq N$
• $u_i\neq v_i$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$u_1$ $v_1$

$\vdots$

$u_M$ $v_M$

Output

Print $N-1$ lines. The $i$-th line should contain the probability, modulo $998244353$, that $G$ is a forest after the $i$-th operation.

3 2
1 2
2 3

1
499122177

Let us denote by $(u, v)$ the edge connecting Vertices $u$ and $v$.

After the $1$-st operation, $G$ will have:

• Edge $(1, 2)$ with a probability of $1/2$;
• Edge $(2, 3)$ with a probability of $1/2$.

In both cases, $G$ is a forest, so the answer for $K=1$ is $1$.

After the $2$-nd operation, $G$ will have:

• Edges $(1, 2)$ and $(1, 2)$ with a probability of $1/4$;
• Edges $(2, 3)$ and $(2, 3)$ with a probability of $1/4$;
• Edges $(1, 2)$ and $(2, 3)$ with a probability of $1/2$.

$G$ is a forest only when $G$ has Edges $(1, 2)$ and $(2, 3)$. Therefore, the sought probability is $1/2$; when represented modulo $998244353$, it is $499122177$, which should be printed.

4 5
1 2
1 2
1 4
2 3
2 4

1
758665709
918384805