Problem Statement

Given is an undirected graph G consisting of N vertices numbered 1 through N and M edges numbered 1 through M. It is guaranteed that G is connected and contains no self-loops and no multi-edges.
Edge i connects Vertex a_i and Vertex b_i bidirectionally.
Each edge can be painted red or blue. Initially, every edge is painted red.
Consider making $a new graph G^{\prime}$ by removing zero or more edges from G.
There are 2^M graphs that $G^{\prime}$ can be. Among them, find the number of good graphs defined below, modulo 998244353.
$G^{\prime}$ is said to be a good graph when both of the following conditions are satisfied:

• $G^{\prime}$ is connected.
• It is possible to make all edges blue by doing the following operation zero or more times.
• Choose one vertex and change the color of every edge incident to that vertex, that is, red to blue and vice versa.

Input

Input is given from Standard Input in the following format:
N M
a_1 b_1
$\vdots$
a_M b_M

Output

Print the number of good graphs among the ones that $G^{\prime}$ can be, modulo 998244353.

Constraints

• All values in input are integers.
• $1 \leq N \leq 17$
• $N-1 \leq M \leq N(N-1)/2$
• G is connected and has no self-loops and no multi-edges.

Sample Input 1

3 2
1 2
2 3

Sample Output 1

1

• The conditions are satisfied only if neither Edge 1 nor Edge 2 is removed.

• In this case, we can make all edges blue by, for example, doing the operation for Vertex 2.

• Otherwise, the graph gets disconnected and thus does not satisfy the condition.

Sample Input 2

4 6
1 2
1 3
1 4
2 3
2 4
3 4

Sample Output 2

19

Sample Input 3

17 50
16 17
10 9
16 10
5 17
6 15
5 9
15 11
16 1
8 13
6 17
15 3
16 15
11 3
7 6
1 4
11 13
10 6
10 12
3 16
7 3
16 5
13 3
12 13
7 11
3 12
13 10
1 12
9 15
11 14
4 6
13 2
6 1
15 2
1 14
15 17
2 11
14 13
16 9
16 8
8 17
17 12
1 11
6 12
17 2
8 1
14 6
9 7
11 10
5 14
17 7

Sample Output 3

90625632

• Be sure to find the count modulo 998244353.

见题面 Input 部分

见题面 Output 部分