Score : $800$ points

Problem Statement

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 $a_j$ ($2 \leq a_j \leq N-1$). 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

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

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 $10^{-9}$.

3
-1 0 2
1 1
2

-1.0
1.0
2.0

Rabbit $2$ will perform the jump. If rabbit $1$ is chosen, the coordinate of the destination will be $-2$. If rabbit $3$ is chosen, the coordinate of the destination will be $4$. Thus, the expected value of the coordinate of the eventual position of rabbit $2$ is $0.5 \times (-2)+0.5 \times 4=1.0$.

3
1 -1 1
2 2
2 2

1.0
-1.0
1.0

$x_i$ may not be distinct.

5
0 1 3 6 10
3 10
2 3 4

0.0
3.0
7.0
8.0
10.0