Score : $1000$ points

Problem Statement

You are given a simple connected undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the $i$-th edge connects vertices $A_i$ and $B_i$.

Find the number of ways to paint each edge of $G$ in color $1$, $2$, or $3$ so that the following condition is satisfied, modulo $998244353$.

• There is a simple path in $G$ that contains an edge in color $1$, an edge in color $2$, and an edge in color $3$.

What is a simple path?

$$(v_1, \ldots, v_{k+1})$$$$(e_1, \ldots, e_k)$$- $i\neq j \implies v_i\neq v_j$; - edge $e_i$ connects vertices $v_i$ and $v_{i+1}$.

## Constraints - $3 \leq N\leq 2\times 10^5$ - $3 \leq M\leq 2\times 10^5$ - $1 \leq A_i, B_i \leq N$ - The given graph is simple and connected. ## Input The 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 ways to paint each edge of $G$ in color $1$, $2$, or $3$ so that the condition in question is satisfied, modulo $998244353$. ```input1 3 3 1 2 1 3 3 2 ``` ```output1 0 ``` Any simple path in $G$ contains two or fewer edges, so there is no way to satisfy the condition. ```input2 4 6 1 2 1 3 1 4 2 3 2 4 3 4 ``` ```output2 534 ``` ```input3 6 5 1 3 4 3 5 4 4 2 1 6 ``` ```output3 144 ``` ```input4 6 7 1 2 2 3 3 1 4 5 5 6 6 4 1 6 ``` ```output4 1794 ```$$