Problem Statement

Given are a tree with N vertices, a sequence of $M numbers A=(A_1,\ldots,A_M)$, and an integer K.
The vertices are numbered 1 through N, and the i-th edge connects Vertices U_i and V_i.
We will paint each of the N-1 edges of this tree red or blue. Among the 2^{N-1} such ways, find the number of ones that satisfies the following condition, modulo 998244353.
Condition:
Let us put a piece on Vertex A_1, and for each $i=1,\ldots,M-1$ in this order, move it from Vertex A_i to Vertex A_{i+1} along the edges in the shortest path. After all of these movements, R-B=K holds, where R and B are the numbers of times the piece traverses a red edge and a blue edge, respectively.

Input

Input is given from Standard Input in the following format:
N M K
$A_1 A_2 \ldots A_M$
U_1 V_1
$\vdots$
U_{N-1} V_{N-1}

Output

Print the answer.

Constraints

• $2 \leq N \leq 1000$
• $2 \leq M \leq 100$
• $|K| \leq 10^5$
• $1 \leq A_i \leq N$
• $1\leq U_i,V_i\leq N$
• The given graph is a tree.
• All values in input are integers.

Sample Input 1

4 5 0
2 3 2 1 4
1 2
2 3
3 4

Sample Output 1

2

If we paint the 1-st and 3-rd edges red and the 2-nd edge blue, the piece will traverse the following numbers of red and blue edges:

• 0 red edges and 1 blue edge when moving from Vertex 2 to 3,
• 0 red edges and 1 blue edge when moving from Vertex 3 to 2,
• 1 red edge and 0 blue edges when moving from Vertex 2 to 1,
• 2 red edges and 1 blue edge when moving from Vertex 1 to 4,

for a total of 3 red edges and 3 blue edges, satisfying the condition.

Another way to satisfy the condition is to paint the 1-st and 3-rd edges blue and the 2-nd edge red. There is no other way to satisfy it, so the answer is 2.

Sample Input 2

3 10 10000
1 2 1 2 1 2 2 1 1 2
1 2
1 3

Sample Output 2

0

There may be no way to paint the tree to satisfy the condition.

Sample Input 3

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

Sample Output 3

126

Sample Input 4

5 8 -1
1 4 1 4 2 1 3 5
1 2
4 1
3 1
1 5

Sample Output 4

2

见题面 Input 部分

见题面 Output 部分