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 Rust (1.15.1) (judge verdict: Accepted)

fn main(){
    let mut sc = Scanner::new();
    while sc.ok(){
        let n = sc.next();
        let mut x = vec![0i64; n];
        for i in 0..n {
            x[i] = sc.next();
        }
        let mut d = vec![0i64; n-1];
        for i in 0..n-1 {
            d[i] = x[i+1] - x[i];
        }
        let m = sc.next();
        let k = sc.next();

        let mut perm:Vec<_> = (0..n-1).collect();
        for _ in 0..m {
            let a = sc.next::<usize>() - 1;
            perm.swap(a-1, a);
        }

        let perm2 = pow(&perm, k);
        let mut res = vec![0i64; n-1];
        for i in 0..n-1 {
            res[i] = d[perm2[i]];
        }

        let mut cur = x[0];
        println!("{}", cur);
        for i in 0..n-1 {
            cur += res[i];
            println!("{}", cur);
        }
    }
}

fn pow(perm: &Vec<usize>, k: usize) -> Vec<usize> {
    if k == 0 {
        (0..perm.len()).collect()
    } else if k%2 == 1 {
        let x = pow(perm, k-1);
        prod(&x, &perm)
    } else {
        let x = pow(perm, k/2);
        prod(&x, &x)
    }
}

fn prod(a: &Vec<usize>, b: &Vec<usize>) -> Vec<usize> {
    let n = a.len();
    let mut res = vec![0; n];
    for i in 0..n {
        res[i] = a[b[i]];
    }
    res
}

#[allow(dead_code)]
struct Scanner {
    buf: std::collections::VecDeque<String>,
}

#[allow(dead_code)]
impl Scanner {
    fn new() -> Scanner {
        Scanner { buf: std::collections::VecDeque::new() }
    }

    fn next<T: std::str::FromStr>(&mut self) -> T {
        self.wrapped::<T>().unwrap()
    }

    fn wrapped<T: std::str::FromStr>(&mut self) -> Option<T> {
        self.read();
        self.buf.pop_front().and_then(|s| s.parse::<T>().ok())
    }

    fn ok(&mut self) -> bool {
        self.read() && !self.buf.is_empty()
    }

    fn eof(&mut self) -> bool {
        !self.ok()
    }

    fn read(&mut self) -> bool {
        while self.buf.is_empty() {
            let mut st = String::new();
            while st.trim() == "" {
                match std::io::stdin().read_line(&mut st) {
                    Ok(l) if l > 0 => continue,
                    _ => return false,
                }
            }
            self.buf = st.split_whitespace().map(|x| x.to_string()).collect();
        }
        true
    }
}