Score : $2000$ points

Problem Statement

You are given a tree with $N$ vertices. The vertices are numbered from $1$ to $N$, and the $i$-th edge connects vertices $u_i$ and $v_i$.

Initially, on each vertex of the tree is written number $1$.

In one operation, you can do the following:

• Choose any vertex $v$ of the tree. Let $X$ be the XOR of the values written in the neighbors of $v$. Then, if the number written on $v$ is $a_v$, replace it by $(a_v\ \mathrm{XOR}\ X)$.

You can perform this operation any finite (maybe zero) number of times. How many configurations can you achieve? As this number can be large, output it modulo $998244353$.

Two configurations are called different if there exists a vertex $v$, such that the numbers written in $v$ in these configurations are different.

Constraints

• $3 \le N \le 2\cdot 10^5$
• $1\le u_i, v_i \le N$
• $u_i \neq v_i$
• The input represents a valid tree.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_{N-1}$ $v_{N-1}$

Output

Output the number of configurations you can get from the initial state, modulo $998244353$.

4
1 2
2 3
3 4

10

We can get the following configurations ($abcd$ means vertices $1,2,3,4$ have $a,b,c,d$ written on them): $1111$, $1110$, $1100$, $1000$, $0111$, $0110$, $0100$, $0011$, $0010$, $0001$.

5
1 2
1 3
1 4
1 5

24

6
1 3
2 3
3 4
4 5
4 6

40

9
1 2
2 3
1 4
4 5
1 6
6 7
7 8
8 9

255