Problem:
Problem C: One-Dimensional Cellular Automaton

There is a one-dimensional cellular automaton consisting of N cells. Cells are numbered from 0 to N − 1.

Each cell has a state represented as a non-negative integer less than M. The states of cells evolve through discrete time steps. We denote the state of the i-th cell at time t as S(i, t). The state at time t + 1 is defined by the equation

S(i, t + 1) = (A × S(i − 1, t) + B × S(i, t) + C × S(i + 1, t)) mod M,         (1)

where A, B and C are non-negative integer constants. For i < 0 or N ≤ i, we define S(i, t) = 0.

Given an automaton definition and initial states of cells, your mission is to write a program that computes the states of the cells at a specified time T.

Input

The input is a sequence of datasets. Each dataset is formatted as follows.

N M A B C T

S(0, 0) S(1, 0) ... S(N − 1, 0)

The first line of a dataset consists of six integers, namely N, M, A, B, C and T. N is the number of cells. M is the modulus in the equation (1). A, B and C are coefficients in the equation (1). Finally, T is the time for which you should compute the states.

You may assume that 0 < N ≤ 50, 0 < M ≤ 1000, 0 ≤ A, B, C < M and 0 ≤ T ≤ 109.

The second line consists of N integers, each of which is non-negative and less than M. They represent the states of the cells at time zero.

A line containing six zeros indicates the end of the input.

Output

For each dataset, output a line that contains the states of the cells at time T. The format of the output is as follows.

S(0, T) S(1, T) ... S(N − 1, T)

Each state must be represented as an integer and the integers must be separated by a space.

Sample Input

5 4 1 3 2 0
0 1 2 0 1
5 7 1 3 2 1
0 1 2 0 1
5 13 1 3 2 11
0 1 2 0 1
5 5 2 0 1 100
0 1 2 0 1
6 6 0 2 3 1000
0 1 2 0 1 4
20 1000 0 2 3 1000000000
0 1 2 0 1 0 1 2 0 1 0 1 2 0 1 0 1 2 0 1
30 2 1 0 1 1000000000
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
30 2 1 1 1 1000000000
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
30 5 2 3 1 1000000000
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0

Output for the Sample Input

0 1 2 0 1
2 0 0 4 3
2 12 10 9 11
3 0 4 2 1
0 4 2 0 4 4
0 376 752 0 376 0 376 752 0 376 0 376 752 0 376 0 376 752 0 376
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 3 2 2 2 3 3 1 4 3 1 2 3 0 4 3 3 0 4 2 2 2 2 1 1 2 1 3 0

Solution in JAVA (judge verdict: Accepted)

import java.util.*;

class Main{

    int n, mod, a, b, c, t;

    void solve(){
        Scanner sc = new Scanner(System.in);

        while(true){
            n = sc.nextInt(); mod = sc.nextInt();
            a = sc.nextInt(); b = sc.nextInt(); c = sc.nextInt();
            t = sc.nextInt();
            if(n==0 && mod==0 && a==0 && b==0 && c==0 && t==0) break;

            long[][] s = new long[n][1];
            for(int i=0; i<n; i++) s[i][0] = sc.nextLong();

            long[][] mat = new long[n][n];
            for(int i=0; i<n; i++){
                for(int j=-1; j<2; j++){
                    int idx = i + j;
                    if(idx<0 || idx>=n) continue;
                    if(j==-1) mat[i][idx] = a;
                    if(j==0) mat[i][idx] = b;
                    if(j==1) mat[i][idx] = c;
                }
            }

            mat = pow(mat, t);
            s = matmul(mat, s);
            for(int i=0; i<s.length-1; i++){
                System.out.print(s[i][0]+" ");
            }
            System.out.println(s[s.length-1][0]);
        }
    }

    //ティツ。ツ古・ツ按療」ツ?ョテヲツ篠崚」ツ?妥ァツョツ療」ツつ津」ツ?凖」ツつ?
    long[][] matmul(long[][] m1, long[][] m2){
        int a = m1.length, b = m2[0].length;
        long[][] res = new long[a][b];
        for(int i=0; i<a; i++){
            for(int j=0; j<b; j++){
                long sum = 0;
                for(int k=0; k<m1[0].length; k++){
                    sum += (m1[i][k] * m2[k][j]) % mod;
                }
                res[i][j] = sum % mod;
            }
        }
        return res;
    }

    //ティツ。ツ古・ツ按療」ツ?ョテァツエツッテ、ツケツ療」ツつ津ィツィツ暗ァツョツ療」ツ?凖」ツつ?
    long[][] pow(long[][] matrix, int l){
        long[][] ret = new long[matrix.length][matrix[0].length];
        for(int i=0; i<matrix.length; i++) ret[i][i] = 1;
        /*
          lテ」ツつ?テゥツ?イテヲツ閉ーテ」ツ?ァティツ。ツィテ」ツ?凖」ツ??
          テ」ツδ禿」ツδε」ツδ暗」ツ?ョ1テ」ツ?古ァツォツ凝」ツ?」テ」ツ?ヲテ」ツ??」ツつ凝」ツ?ィテ」ツ?禿」ツつ催」ツ??」ツ?妥ヲツ篠崚」ツ?妥」ツつ凝」ツ??
         テッツシツ暗、ツセツ凝ッツシツ瑛=19テ」ツ?ョテ」ツ?ィテ」ツ??
          l = 10011
          A^19 = A^1 * A^2 * A^16
        */
        for(; l>0; l>>=1){
            if((l&1)>0) ret = matmul(ret, matrix);
            matrix = matmul(matrix, matrix);
        }
        return ret;
    }

    public static void main(String[] args){
        new Main().solve();
    }
}