Problem: 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

Solution (in CPP):

Solution:
// luogu-judger-enable-o2
#include<bits/stdc++.h>
#define maxn 100010
#define LL long long
using namespace std;
LL n,a[maxn],y,id[maxn],vis[maxn],stk[maxn],top,k,m,ans[maxn];
int main(){
	scanf("%lld",&n);
	for(LL i=1;i<=n;i++){
		scanf("%lld",&a[i]);
		id[i]=i;
	}
	for(LL i=n;i>=1;i--){//做差分 
		a[i]-=a[i-1];
	}
	scanf("%lld%lld",&m,&k);
	for(LL i=1;i<=m;i++){
		scanf("%lld",&y);
		swap(id[y],id[y+1]);//（难点）归纳出来的规律 
	}
	for(LL i=1;i<=n;i++){
		if(vis[i]==0){
			top=0;
			for(LL j=i;!vis[j];j=id[j]){
				stk[++top]=j;
				vis[j]=1;
			}
			for(LL j=1;j<=top;j++){
				ans[stk[j]]=a[stk[(k+j-1)%top+1]];
			}
		}
	}
	for(LL i=1;i<=n;i++){
		ans[i]+=ans[i-1];//期望值的可加性 
		printf("%lld\n",ans[i]);
	}
	return 0;
}//a b c d e f