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Contest: Task: Related: TaskD TaskF

Score : $500$ points

Problem Statement

You are given a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1, \dots, N$, and the $i$-th $(1 \leq i \leq M)$ edge connects Vertices $U_i$ and $V_i$.

There are $2^N$ ways to paint each vertex red or blue. Find the number, modulo $998244353$, of such ways that satisfy all of the following conditions:

There are exactly $K$ vertices painted red.
There is an even number of edges connecting vertices painted in different colors.

Constraints

$2 \leq N \leq 2 \times 10^5$
$1 \leq M \leq 2 \times 10^5$
$0 \leq K \leq N$
$1 \leq U_i \lt V_i \leq N \, (1 \leq i \leq M)$
$(U_i, V_i) \neq (U_j, V_j) \, (i \neq j)$
All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$
$U_1$ $V_1$
$\vdots$
$U_M$ $V_M$

Output

Print the answer.

Sample Input 1

Sample Output 1

The following two ways satisfy the conditions.

Paint Vertices $1$ and $2$ red and Vertices $3$ and $4$ blue.
Paint Vertices $3$ and $4$ red and Vertices $1$ and $2$ blue.

In either of the ways above, the $2$-nd and $3$-rd edges connect vertices painted in different colors.

Sample Input 2