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;//リストの使用
using System.Collections.Generic;
using System.Text;//テキストの高速出力に必要
class Program
{
	static void Main()
	{
		string[] input = Console.ReadLine().Split(' ');
		int n = int.Parse(input[0]);
  	int m = int.Parse(input[1]);
    List<long>[] switchList = new List<long>[m];
    for(int i = 0; i < m; i++)
    {
      switchList[i] = new List<long>();
      long[] nums = Array.ConvertAll(Console.ReadLine().Split(' '),long.Parse);
      for(int j = 0; j < nums[0]; j++)
      {
        switchList[i].Add(nums[j+1]);
      }
    }
    long[] aims = Array.ConvertAll(Console.ReadLine().Split(' '),long.Parse);
		long answer = 0;

    for(long i = 0; i < (Math.Pow(2,n)); i++)//２進数で全通り試す
      {
        long ii = i;
        int[] lightMemo = new int[m];
        for(int k = 0; k < m; k++) lightMemo[k] = 0;
        
        for(long j = 0; j < n; j++)//全ての桁
        {
          if(ii%2==0)
          {
          for(int k = 0; k < m; k++)
          {
            for(int l = 0; l < switchList[k].Count(); l++)
            {
              //Console.WriteLine(switchList[k][l]);
              if(switchList[k][l] == j+1)
              {
                lightMemo[k]++;
                lightMemo[k] %= 2;
                break;
              }
            }
          }
          }
          
          ii /= 2;
        }

        for(int k = 0; k < m; k++)
        {
          //Console.WriteLine(lightMemo[k]);
          if(lightMemo[k] != aims[k]) break;
          if(k == m-1) answer++;
        }
        
      }
    
		Console.WriteLine(answer);
	}
}