path pass i

We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i. Additionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.

For each k=1, 2, ..., N, solve the following problem:

• Find the number of simple paths that visit a vertex painted in the color k one or more times.

Note: The simple paths from Vertex u to v and from v to u are not distinguished.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq c_i \leq N
• 1 \leq a_i,b_i \leq N
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

N c_1 c_2 ... c_N a_1 b_1 : a_{N-1} b_{N-1}

Output

Print the answers for k = 1, 2, ..., N in order, each in its own line.

Examples

Input

3 1 2 1 1 2 2 3

Output

5 4 0

Input

1 1

Output

1

Input

2 1 2 1 2

Output

2 2

Input

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

Output

5 8 10 5 5

Input

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

Output

18 15 0 14 23 0 23 0

inputFormat

input are integers.

Input

Input is given from Standard Input in the following format:

N c_1 c_2 ... c_N a_1 b_1 : a_{N-1} b_{N-1}

outputFormat

Output

Print the answers for k = 1, 2, ..., N in order, each in its own line.

Examples

Input

3 1 2 1 1 2 2 3

Output

5 4 0

Input

1 1

Output

1

Input

2 1 2 1 2

Output

2 2

Input

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

Output

5 8 10 5 5

Input

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

Output

18 15 0 14 23 0 23 0

样例

3
1 2 1
1 2
2 3

5
4
0

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