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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Text;

public class AtCoderC
{
    private static int n;
    private static int m;
    private static int[,] lights;
    private static int[] conditions;
    static void Main()
    {
        {
            string line = Console.ReadLine();
            var words = line.Split(' ');
            n = int.Parse(words[0]);
            m = int.Parse(words[1]);
        }

        lights = new int[m, n];
        {
            for (int i = 0; i < m; i++)
            {
                string line = Console.ReadLine();
                var words = line.Split(' ');
                int num = int.Parse(words[0]);
                for (int j = 0; j < num; j++)
                {
                    int sw = int.Parse(words[j + 1]);
                    lights[i, sw - 1] = 1;
                }
            }
        }

        conditions = new int[m];
        {
            string line = Console.ReadLine();
            var words = line.Split(' ');
            for (int i = 0; i < m; i++)
            {
                conditions[i] = int.Parse(words[i]);
            }
        }

        int[] array = new int[n];
        int answer = Progress(array, 0);

        Console.WriteLine(answer);
    }

    private static int Progress(int[] array, int index)
    {
        if (index == n)
        {
            for (int i = 0; i < m; i++)
            {
                int count = 0;
                for (int j = 0; j < n; j++)
                {
                    if (array[j] == 1)
                    {
                        if (lights[i, j] == 1)
                        {
                            count++;
                        }
                    }
                }
                if (count % 2 != conditions[i])
                {
                    return 0;
                }
            }
            return 1;
        }
        int[] newArray = (int[])array.Clone();
        newArray[index] = 1;
        return Progress(array, index + 1) + Progress(newArray, index + 1);
    }
}