Problem:
Score : 300 points

Problem Statement

We have N switches with "on" and "off" state, and M bulbs. The switches are numbered 1 to N, and the bulbs are numbered 1 to M.

Bulb i is connected to k_i switches: Switch s_{i1}, s_{i2}, ..., and s_{ik_i}. It is lighted when the number of switches that are "on" among these switches is congruent to p_i modulo 2.

How many combinations of "on" and "off" states of the switches light all the bulbs?

Constraints

1 \leq N, M \leq 10

1 \leq k_i \leq N

1 \leq s_{ij} \leq N

s_{ia} \neq s_{ib} (a \neq b)

p_i is 0 or 1.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M
k_1 s_{11} s_{12} ... s_{1k_1}
:
k_M s_{M1} s_{M2} ... s_{Mk_M}
p_1 p_2 ... p_M

Output

Print the number of combinations of "on" and "off" states of the switches that light all the bulbs.

Sample Input 1

2 2
2 1 2
1 2
0 1

Sample Output 1

1

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

There are four possible combinations of states of (Switch 1, Switch 2): (on, on), (on, off), (off, on) and (off, off). Among them, only (on, on) lights all the bulbs, so we should print 1.

Sample Input 2

2 3
2 1 2
1 1
1 2
0 0 1

Sample Output 2

0

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an even number of switches that are "on" among the following: Switch 1.

Bulb 3 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

Switch 1 has to be "off" to light Bulb 2 and Switch 2 has to be "on" to light Bulb 3, but then Bulb 1 will not be lighted. Thus, there are no combinations of states of the switches that light all the bulbs, so we should print 0.

Sample Input 3

5 2
3 1 2 5
2 2 3
1 0

Sample Output 3

8

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.List;

public class Main {
    static void solve(MyScanner in, PrintWriter out) {
        int n = in.nextInt();
        int m = in.nextInt();

        List<List<Integer>> s = new ArrayList<>(m);
        for (int i = 0; i < m; i++) {
            int k = in.nextInt();
            s.add(new ArrayList<>(k));
            for (int j = 0; j < k; j++) {
                s.get(i).add(in.nextInt());
            }
        }

        int[] p = in.nextIntArray(m);

        int ans = 0;
        label : for (int bit = 0; bit < (1 << n); bit++) {
            for (int i = 0; i < s.size(); i++) {
                int c = 0;
                List<Integer> l = s.get(i);
                for (int v : l) {
                    if ((bit & (1 << (v - 1))) != 0) {
                        c++;
                    }
                }
                if (c % 2 != p[i])
                    continue label;
            }
            ans++;
        }
        out.println(ans);
    }

    public static void main(String[] args) {
        PrintWriter w = new PrintWriter(System.out);
        solve(new MyScanner(System.in), w);
        w.flush();
    }

    static final class MyScanner {
        static final int BUFFER_SIZE = 8192;
        private final InputStream in;
        private final byte[] buffer = new byte[BUFFER_SIZE];
        private int point;
        private int readLength;
        MyScanner(InputStream in) {
            this.in = in;
        }
        private int readByte() {
            if (point < readLength) {
                int result = buffer[point];
                point += 1;
                return result;
            }
            try {
                readLength = in.read(buffer);
            } catch (IOException e) {
                throw new AssertionError(null, e);
            }
            if (readLength == -1) {
                return -1;
            }
            point = 1;
            return buffer[0];
        }
        private static boolean isPrintableCharExceptSpace(int c) {
            return 33 <= c && c <= 126;
        }
        String next() {
            int c = readByte();
            while (!(c == -1 || isPrintableCharExceptSpace(c))) {
                c = readByte();
            }
            assert c != -1;
            StringBuilder b = new StringBuilder();
            do {
                b.appendCodePoint(c);
                c = readByte();
            } while (c != -1 && isPrintableCharExceptSpace(c));
            return b.toString();
        }
        long nextLong() {
            int c = readByte();
            while (!(c == -1 || isPrintableCharExceptSpace(c))) {
                c = readByte();
            }
            assert c != -1;
            boolean minus = false;
            if (c == '-') {
                minus = true;
                c = readByte();
            }
            long result = 0L;
            do {
                assert '0' <= c && c <= '9' : c;
                result = result * 10L + (c - '0');
                c = readByte();
            } while (c != -1 && isPrintableCharExceptSpace(c));
            return minus ? -result : result;
        }
        int nextInt() {
            long n = nextLong();
            assert Integer.MIN_VALUE <= n && n <= Integer.MAX_VALUE : n;
            return (int)n;
        }
        double nextDouble() {
            return Double.parseDouble(next());
        }
        int[] nextIntArray(int n) {
            int[] result = new int[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextInt();
            }
            return result;
        }
        private static boolean allSameLength(int[] a, int[] b, int[]... c) {
            if (a.length != b.length) {
                return false;
            }
            for (int i = 0; i < c.length; i++) {
                if (a.length != c[i].length) {
                    return false;
                }
            }
            return true;
        }
        private static boolean allSameLength(char[] a, char[] b, char[]... c) {
            if (a.length != b.length) {
                return false;
            }
            for (int i = 0; i < c.length; i++) {
                if (a.length != c[i].length) {
                    return false;
                }
            }
            return true;
        }
        void nextVerticalIntArrays(int[] a, int[] b, int[]... c) {
            assert allSameLength(a, b, c);
            for (int i = 0; i < a.length; i++) {
                a[i] = nextInt();
                b[i] = nextInt();
                for (int[] d : c) {
                    d[i] = nextInt();
                }
            }
        }
        long[] nextLongArray(int n) {
            long[] result = new long[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextLong();
            }
            return result;
        }
        char nextChar() {
            int c = readByte();
            while (!(c == -1 || isPrintableCharExceptSpace(c))) {
                c = readByte();
            }
            assert c != -1;
            return (char)c;
        }
        char[] nextCharArray(int n) {
            char[] result = new char[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextChar();
            }
            return result;
        }
        char[][] next2dCharArray(int n, int m) {
            char[][] result = new char[n][m];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < m; j++) {
                    result[i][j] = nextChar();
                }
            }
            return result;
        }
        void nextVerticalCharArrays(char[] a, char[] b, char[]... c) {
            assert allSameLength(a, b, c);
            for (int i = 0; i < a.length; i++) {
                a[i] = nextChar();
                b[i] = nextChar();
                for (char[] d : c) {
                    d[i] = nextChar();
                }
            }
        }
    }
}