Distributing Integers

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

• Consider writing a number on each vertex in the tree in the following manner:
• First, write 1 on Vertex k.
• Then, for each of the numbers 2, ..., N in this order, write the number on the vertex chosen as follows:
• Choose a vertex that still does not have a number written on it and is adjacent to a vertex with a number already written on it. If there are multiple such vertices, choose one of them at random.
• Find the number of ways in which we can write the numbers on the vertices, modulo (10^9+7).

Constraints

• 2 \leq N \leq 2 \times 10^5
• 1 \leq a_i,b_i \leq N
• The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N a_1 b_1 : a_{N-1} b_{N-1}

Output

For each k=1, 2, ..., N in this order, print a line containing the answer to the problem.

Examples

Input

3 1 2 1 3

Output

2 1 1

Input

2 1 2

Output

1 1

Input

5 1 2 2 3 3 4 3 5

Output

2 8 12 3 3

Input

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

Output

40 280 840 120 120 504 72 72

inputFormat

Input

Input is given from Standard Input in the following format:

N a_1 b_1 : a_{N-1} b_{N-1}

outputFormat

Output

For each k=1, 2, ..., N in this order, print a line containing the answer to the problem.

Examples

Input

3 1 2 1 3

Output

2 1 1

Input

2 1 2

Output

1 1

Input

5 1 2 2 3 3 4 3 5

Output

2 8 12 3 3

Input

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

Output

40 280 840 120 120 504 72 72

样例

3
1 2
1 3

2
1
1

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