Problem:
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≤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}.

Sample Input 1

3
-1 0 2
1 1
2

Sample Output 1

-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×(-2)+0.5×4=1.0.

Sample Input 2

3
1 -1 1
2 2
2 2

Sample Output 2

1.0
-1.0
1.0

x_i may not be distinct.

Sample Input 3

5
0 1 3 6 10
3 10
2 3 4

Sample Output 3

0.0
3.0
7.0
8.0
10.0

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <vector>
#include <numeric>
#define repeat(i,n) for (int i = 0; (i) < (n); ++(i))
#define whole(f,x,...) ([&](decltype((x)) whole) { return (f)(begin(whole), end(whole), ## __VA_ARGS__); })(x)
typedef long long ll;
using namespace std;
template <typename T>
vector<T> apply(vector<T> const & a, vector<int> const & swp) {
    int n = swp.size();
    vector<T> b(n);
    repeat (i,n) b[i] = a[swp[i]];
    return b;
}
int main() {
    int n; cin >> n;
    vector<ll> x(n); repeat (i,n) cin >> x[i];
    int m; ll k; cin >> m >> k;
    vector<int> a(m); repeat (i,m) { cin >> a[i]; -- a[i]; }
    vector<ll> dx(n-1); repeat (i,n-1) dx[i] = x[i+1] - x[i];
    vector<int> swp(n-1); whole(iota, swp, 0);
    repeat (j,m) swap(swp[a[j]-1], swp[a[j]]);
    for (ll i = 1; i <= k; i <<= 1) {
        if (k & i) dx = apply(dx, swp);
        swp = apply(swp, swp);
    }
    vector<ll> y(n);
    y[0] = x[0]; repeat (i,n-1) y[i+1] = y[i] + dx[i];
    repeat (i,n) cout << y[i] << endl;
    return 0;
}