AtCoder Grand Contest 006 - 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

Difficulty: UNKNOWN_DIFFICULTY

Starter code:

Solutions:
n = int(input())
x = list(map(int, input().split()))
(m, k) = map(int, input().split())
a = list(map(int, input().split()))
b = []
x2 = []
c = []
for i in range(1, n):
	b.append(i - 1)
	x2.append(x[i] - x[i - 1])
for i in range(m):
	to = a[i] - 1
	(b[to], b[to - 1]) = (b[to - 1], b[to])
c = [-1] * (n - 1)
for i in range(n - 1):
	c[b[i]] = i
for i in range(60):
	x3 = [0] * (n - 1)
	if k & 1 << i:
		for i in range(n - 1):
			x3[c[i]] = x2[i]
		x2 = x3
	c2 = [-1] * (n - 1)
	for i in range(n - 1):
		c2[i] = c[c[i]]
	c = c2
s = x[0]
print(s)
for i in x2:
	s += i
	print(s)


n = int(input())
x = list(map(int, input().split()))
g = [x[i + 1] - x[i] for i in range(n - 1)]
(m, k) = map(int, input().split())
A = list(map(int, input().split()))
t = list(range(n - 1))
for a in A:
	(t[a - 2], t[a - 1]) = (t[a - 1], t[a - 2])
s = list(range(n - 1))
while k > 0:
	if k & 1:
		s = [s[t[i]] for i in range(n - 1)]
	t = [t[t[i]] for i in range(n - 1)]
	k >>= 1
new_g = [g[s[i]] for i in range(n - 1)]
ans = [x[0]]
for i in range(n - 1):
	ans.append(ans[-1] + new_g[i])
print(*ans, sep='\n')


import sys
readline = sys.stdin.readline

def merge(A, B):
	N = len(A)
	res = [B[A[i]] for i in range(N)]
	return res
N = int(readline())
X = list(map(int, readline().split()))
(M, K) = map(int, readline().split())
A = list(map(lambda x: int(x) - 1, readline().split()))
Y = [x2 - x1 for (x1, x2) in zip(X, X[1:])]
P = list(range(N))
for a in A:
	(P[a], P[a + 1]) = (P[a + 1], P[a])
stack = [P]
for _ in range(65):
	stack.append(merge(stack[-1], stack[-1]))
E = list(range(N))
for i in range(K.bit_length()):
	if 1 << i & K:
		E = merge(E, stack[i])
Ans = [0] * N
Ans[0] = X[0]
for i in range(1, N):
	Ans[i] = Ans[i - 1] + Y[E[i] - 1]
print('\n'.join(map(str, Ans)))


N = int(input())
X = [int(i) for i in input().split()]
dX = []
for i in range(N - 1):
	dX.append(X[i + 1] - X[i])
(M, K) = map(int, input().split())
A = [int(i) - 2 for i in input().split()]
Swap = [list(range(N - 1)) for i in range(61)]
for a in A:
	(Swap[0][a], Swap[0][a + 1]) = (Swap[0][a + 1], Swap[0][a])
for k in range(60):
	for i in range(N - 1):
		Swap[k + 1][i] = Swap[k][Swap[k][i]]
C = list(range(N - 1))
for k in range(61)[::-1]:
	if K & 1 << k:
		for i in range(N - 1):
			C[i] = Swap[k][C[i]]
new_dX = [0] * (N - 1)
for i in range(N - 1):
	new_dX[i] = dX[C[i]]
S = X[0]
print(X[0])
for i in range(N - 1):
	S += new_dX[i]
	print(S)