Rabbit Exercise

There are N rabbits on a number line. The rabbits are conveniently numbered 1 through N. The coordinate of the initial position of rabbit i is x_i.

The rabbits will now take exercise on the number line, by performing sets described below. A set consists of M jumps. The j-th jump of a set is performed by rabbit a_j (2≤a_j≤N-1). For this jump, either rabbit a_j-1 or rabbit a_j+1 is chosen with equal probability (let the chosen rabbit be rabbit x), then rabbit a_j will jump to the symmetric point of its current position with respect to rabbit x.

The rabbits will perform K sets in succession. For each rabbit, find the expected value of the coordinate of its eventual position after K sets are performed.

Constraints

• 3≤N≤10^5
• x_i is an integer.
• |x_i|≤10^9
• 1≤M≤10^5
• 2≤a_j≤N-1
• 1≤K≤10^{18}

Input

The input is given from Standard Input in the following format:

N x_1 x_2 ... x_N M K a_1 a_2 ... a_M

Output

Print N lines. The i-th line should contain the expected value of the coordinate of the eventual position of rabbit i after K sets are performed. The output is considered correct if the absolute or relative error is at most 10^{-9}.

Examples

Input

3 -1 0 2 1 1 2

Output

-1.0 1.0 2.0

Input

3 1 -1 1 2 2 2 2

Output

1.0 -1.0 1.0

Input

5 0 1 3 6 10 3 10 2 3 4

Output

0.0 3.0 7.0 8.0 10.0

inputFormat

Input

The input is given from Standard Input in the following format:

N x_1 x_2 ... x_N M K a_1 a_2 ... a_M

outputFormat

Output

Print N lines. The i-th line should contain the expected value of the coordinate of the eventual position of rabbit i after K sets are performed. The output is considered correct if the absolute or relative error is at most 10^{-9}.

Examples

Input

3 -1 0 2 1 1 2

Output

-1.0 1.0 2.0

Input

3 1 -1 1 2 2 2 2

Output

1.0 -1.0 1.0

Input

5 0 1 3 6 10 3 10 2 3 4

Output

0.0 3.0 7.0 8.0 10.0

样例

3
1 -1 1
2 2
2 2

1.0
-1.0
1.0

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