Problem Description

ckw has an undirected tree. He will randomly select a node and start a random walk, where each unit of time he jumps to any node within a distance of at most $2$ from the current node with equal probability. There are some nodes marked on the tree. Calculate the expected time for ckw to first reach a marked node.

Input Format

The first line contains two numbers $n$ and $m$, representing the number of nodes and the number of marked nodes, respectively.

The next $n-1$ lines each contain two numbers $x$ and $y$, indicating that there is an edge between nodes $x$ and $y$.

The next $m$ lines each contain a number representing a marked node (may have duplicates).

Output Format

Output $n$ lines, each containing a number. The $i$-th line represents the expected number of steps starting from node $i$, modulo $998244353$.

Sample Input and Output

Sample Input #1

2 1
1 2
1

Sample Output #1

0
2

Sample Input #2

10 2
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9 10
2
5

Sample Output #2

3
0
3
4
0
8
10
13
14
15

Sample Input #3

10 2
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
1 10
3
6

Sample Output #3

5
5
0
5
5
0
5
5
5
5

Notes

$2 \le n \le 10^5, 1 \le m \le n$

$subtask1(20pts): n \le 300$

$subtask2(16pts):$ The $i$-th edge connects nodes $i$ and $i+1$

$subtask3(8pts):$ The $i$-th edge connects nodes $1$ and $i+1$

$subtask4(20pts): n \le 3000$, and the maximum degree of any node is at most $4$

$subtask5(36pts):$ No additional restrictions

见题面 Input 部分

见题面 Output 部分