Problem: Due to the recent popularity of the Deep learning new countries are starting to look like Neural Networks. That is, the countries are being built deep with many layers, each layer possibly having many cities. They also have one entry, and one exit point.

There are exactly L layers, each having N cities. Let us look at the two adjacent layers L1 and L2. Each city from the layer L1 is connected to each city from the layer L2 with the traveling cost cij for <image>, and each pair of adjacent layers has the same cost in between their cities as any other pair (they just stacked the same layers, as usual). Also, the traveling costs to each city from the layer L2 are same for all cities in the L1, that is cij is the same for <image>, and fixed j.

Doctor G. needs to speed up his computations for this country so he asks you to find the number of paths he can take from entry to exit point such that his traveling cost is divisible by given number M.

Input

The first line of input contains N (1 ≤ N ≤ 106), L (2 ≤ L ≤ 105) and M (2 ≤ M ≤ 100), the number of cities in each layer, the number of layers and the number that travelling cost should be divisible by, respectively.

Second, third and fourth line contain N integers each denoting costs 0 ≤ cost ≤ M from entry point to the first layer, costs between adjacent layers as described above, and costs from the last layer to the exit point.

Output

Output a single integer, the number of paths Doctor G. can take which have total cost divisible by M, modulo 109 + 7.

Example

Input

2 3 13
4 6
2 1
3 4


Output

2

Note

<image>

This is a country with 3 layers, each layer having 2 cities. Paths <image>, and <image> are the only paths having total cost divisible by 13. Notice that input edges for layer cities have the same cost, and that they are same for all layers.

Solution (in JAVA):

Solution:
import java.util.*;
import java.io.*;
import java.math.*;
import java.lang.*;

public class Main {
    private static FastReader sc = new FastReader(System.in);
    private static OutputWriter out = new OutputWriter(System.out);
    private static long MOD = (int) (1e9 + 7);
    private static int m;

    public static void main(String[] args) throws Exception {
        int n = sc.nextInt();
        int l = sc.nextInt();
        m = sc.nextInt();
        long[] arr = new long[m];
        for (int i = 0; i < n; i++) {
            int x = sc.nextInt();
            arr[x % m]++;
        }
        HashMap<Integer, Integer> map = new HashMap<>();
        long[] levChange = new long[n];
        for (int i = 0; i < m; i++) map.put(i, 0);
        for (int i = 0; i < n; i++) {
            int x = sc.nextInt();
            levChange[i] = x;
            map.put(x % m, map.get(x % m) + 1);
        }

        long[][] exp = matrixExp(map, l - 2);

        long[] secLast = new long[m];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                secLast[i] = (secLast[i] + ((arr[j] * exp[j][i]) % MOD)) % MOD;
            }
        }

        long ans = 0;
        for (int i = 0; i < n; i++) {
            long x = sc.nextInt();
            ans = (ans + secLast[(int) ((3 * m - levChange[i] - x) % m)]) % MOD;
        }
        System.out.println(ans);
    }

    private static long[][] matrixExp(HashMap<Integer, Integer> map, int b) {
        long[][] mat = new long[m][m];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                mat[i][j] = map.get((j - i + m) % m);
            }
        }

        long[][] res = new long[m][m];
        for (int i = 0; i < m; i++) res[i][i] = 1;
        while (b >= 1) {
            if (b % 2 == 1) {
                res = exp(res, mat);
            }
            b = (b / 2);
            mat = exp(mat, mat);
        }
        return res;
    }

    private static long[][] exp(long[][] a, long[][] b) {
        long[][] ans = new long[m][m];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                for (int k = 0; k < m; k++) {
                    ans[i][j] = (ans[i][j] + ((a[i][k] * b[k][j]) % MOD)) % MOD;
                }
            }
        }
        return ans;
    }
}

class FastReader {
    private boolean finished = false;

    private InputStream stream;
    private byte[] buf = new byte[1024];
    private int curChar;
    private int numChars;
    private SpaceCharFilter filter;

    public FastReader(InputStream stream) {
        this.stream = stream;
    }

    public int read() {
        if (numChars == -1) {
            throw new InputMismatchException();
        }
        if (curChar >= numChars) {
            curChar = 0;
            try {
                numChars = stream.read(buf);
            } catch (IOException e) {
                throw new InputMismatchException();
            }
            if (numChars <= 0) {
                return -1;
            }
        }

        return buf[curChar++];
    }

    public int peek() {
        if (numChars == -1) {
            return -1;
        }
        if (curChar >= numChars) {
            curChar = 0;
            try {
                numChars = stream.read(buf);
            } catch (IOException e) {
                return -1;
            }
            if (numChars <= 0) {
                return -1;
            }
        }
        return buf[curChar];
    }

    public int nextInt() {
        int c = read();
        while (isSpaceChar(c))
            c = read();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = read();
        }
        int res = 0;
        do {
            if (c == ',') {
                c = read();
            }
            if (c < '0' || c > '9') {
                throw new InputMismatchException();
            }
            res *= 10;
            res += c - '0';
            c = read();
        } while (!isSpaceChar(c));
        return res * sgn;
    }

    public long nextLong() {
        int c = read();
        while (isSpaceChar(c))
            c = read();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = read();
        }
        long res = 0;
        do {
            if (c < '0' || c > '9') {
                throw new InputMismatchException();
            }
            res *= 10;
            res += c - '0';
            c = read();
        } while (!isSpaceChar(c));
        return res * sgn;
    }

    public String nextString() {
        int c = read();
        while (isSpaceChar(c))
            c = read();
        StringBuilder res = new StringBuilder();
        do {
            res.appendCodePoint(c);
            c = read();
        } while (!isSpaceChar(c));
        return res.toString();
    }

    public boolean isSpaceChar(int c) {
        if (filter != null) {
            return filter.isSpaceChar(c);
        }
        return isWhitespace(c);
    }

    public static boolean isWhitespace(int c) {
        return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
    }

    private String readLine0() {
        StringBuilder buf = new StringBuilder();
        int c = read();
        while (c != '\n' && c != -1) {
            if (c != '\r') {
                buf.appendCodePoint(c);
            }
            c = read();
        }
        return buf.toString();
    }

    public String nextLine() {
        String s = readLine0();
        while (s.trim().length() == 0)
            s = readLine0();
        return s;
    }

    public String nextLine(boolean ignoreEmptyLines) {
        if (ignoreEmptyLines) {
            return nextLine();
        } else {
            return readLine0();
        }
    }

    public BigInteger nextBigInteger() {
        try {
            return new BigInteger(nextString());
        } catch (NumberFormatException e) {
            throw new InputMismatchException();
        }
    }

    public char nextCharacter() {
        int c = read();
        while (isSpaceChar(c))
            c = read();
        return (char) c;
    }

    public double nextDouble() {
        int c = read();
        while (isSpaceChar(c))
            c = read();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = read();
        }
        double res = 0;
        while (!isSpaceChar(c) && c != '.') {
            if (c == 'e' || c == 'E') {
                return res * Math.pow(10, nextInt());
            }
            if (c < '0' || c > '9') {
                throw new InputMismatchException();
            }
            res *= 10;
            res += c - '0';
            c = read();
        }
        if (c == '.') {
            c = read();
            double m = 1;
            while (!isSpaceChar(c)) {
                if (c == 'e' || c == 'E') {
                    return res * Math.pow(10, nextInt());
                }
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                m /= 10;
                res += (c - '0') * m;
                c = read();
            }
        }
        return res * sgn;
    }

    public boolean isExhausted() {
        int value;
        while (isSpaceChar(value = peek()) && value != -1)
            read();
        return value == -1;
    }

    public String next() {
        return nextString();
    }

    public SpaceCharFilter getFilter() {
        return filter;
    }

    public void setFilter(SpaceCharFilter filter) {
        this.filter = filter;
    }

    public interface SpaceCharFilter {
        public boolean isSpaceChar(int ch);
    }
}

class OutputWriter {
    private final PrintWriter writer;

    public OutputWriter(OutputStream outputStream) {
        writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(outputStream)));
    }

    public OutputWriter(Writer writer) {
        this.writer = new PrintWriter(writer);
    }

    public void print(Object... objects) {
        for (int i = 0; i < objects.length; i++) {
            if (i != 0)
                writer.print(' ');
            writer.print(objects[i]);
        }
    }

    public void printLine(Object... objects) {
        print(objects);
        writer.println();
    }

    public void close() {
        writer.close();
    }

    public void flush() {
        writer.flush();
    }

}