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.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Scanner;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        Scanner in = new Scanner(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        OneDimensionalCellularAutomaton solver = new OneDimensionalCellularAutomaton();
        try {
            int testNumber = 1;
            while (true)
                solver.solve(testNumber++, in, out);
        } catch (UnknownError e) {
            out.close();
        }
    }

    static class OneDimensionalCellularAutomaton {
        public void solve(int testNumber, Scanner in, PrintWriter out) {
            int N = in.nextInt();
            int M = in.nextInt();
            int A = in.nextInt();
            int B = in.nextInt();
            int C = in.nextInt();
            int T = in.nextInt();
            if (N == 0) throw new UnknownError();
            Matrix a = new Matrix(N, N, M);
            for (int i = 0; i < N; i++) {
                a.setCell(i, i, B);
                if (0 < i) {
                    a.setCell(i, i - 1, A);
                }
                if (i < N - 1) {
                    a.setCell(i, i + 1, C);
                }
            }
            Matrix na = Matrix.pow(a, T);
            Matrix s = new Matrix(N, 1, M);
            for (int i = 0; i < N; i++) {
                s.setCell(i, 0, in.nextInt());
            }
            long[][] ans = Matrix.mul(na, s).getArray();
            for (int i = 0; i < N; i++) {
                out.print(ans[i][0]);
                if (i + 1 == N) {
                    out.println();
                } else {
                    out.print(" ");
                }
            }
        }

    }

    static class Matrix {
        private int MOD;
        private long[][] mat;

        public Matrix(int R, int C, int mod) {
            this.mat = new long[R][C];
            this.MOD = mod;
        }

        public Matrix(int R, int C) {
            this.mat = new long[R][C];
            this.MOD = -1;
        }

        public void setCell(int r, int c, long val) {
            if (this.MOD != -1) {
                val %= MOD;
            }
            this.mat[r][c] = val;
        }

        public void addCell(int r, int c, long val) {
            if (this.MOD != -1) {
                val %= MOD;
            }
            this.mat[r][c] += val;
            this.mat[r][c] %= MOD;
        }

        public long[][] getArray() {
            return this.mat.clone();
        }

        public int getRow() {
            return this.mat.length;
        }

        public int getCol() {
            return this.mat[0].length;
        }


        protected Object clone() {
            Matrix ret = new Matrix(mat.length, mat[0].length, MOD);
            ret.mat = this.mat.clone();
            return ret;
        }

        public static Matrix mul(Matrix a, Matrix b) {
            if (a.MOD != b.MOD) throw new IllegalArgumentException();
            if (a.getCol() != b.getRow()) throw new IllegalArgumentException();
            Matrix ret = new Matrix(a.getRow(), b.getCol(), a.MOD);
            for (int i = 0; i < ret.getRow(); i++) {
                for (int j = 0; j < ret.getCol(); j++) {
                    for (int k = 0; k < a.getCol(); k++) {
                        ret.addCell(i, j, a.mat[i][k] * b.mat[k][j]);
                    }
                }
            }
            return ret;
        }

        public static Matrix getIdentityMatrix(int rc, int mod) {
            Matrix ret = new Matrix(rc, rc, mod);
            for (int i = 0; i < rc; i++) {
                ret.setCell(i, i, 1);
            }
            return ret;
        }

        public static Matrix pow(Matrix a, long p) {
            if (a.getCol() != a.getRow()) {
                throw new IllegalArgumentException();
            }
            if (p == 0) {
                return Matrix.getIdentityMatrix(a.getRow(), a.MOD);
            }
            Matrix ret = (Matrix) a.clone();
            if (p % 2 == 0) {
                Matrix t = pow(a, p / 2);
                return mul(t, t);
            } else {
                return mul(a, pow(a, p - 1));
            }
        }

    }
}