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.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Security.Principal;

class Switches
{
    static void Main(string[] args)
    {
        var input = Console.ReadLine().Split(' ');

        var switches = int.Parse(input[0]);
        var number = int.Parse(input[1]);
        var ans = 0;

        int[][] list = new int[number][];
        int[] switchOn = new int[number];

        for(var i = 0; i < number; i++)
        {
            var info = Console.ReadLine().Split(' ');
            
            list[i] = info.Select(int.Parse).ToArray();

        }

        var checkRemaind = Console.ReadLine().Split(' ');
        bool flag = true;

        for (var i = 0; i < Math.Pow(2, switches); i++)
        {
            for(var j = 0; j < switches; j++)
            {
                if((i >> j & 1)== 1)
                {
                    for(var k = 0;k < number; k++)
                    {
                        for(var l = 1;l <= list[k][0];l++)
                        {
                            if((j + 1) == list[k][l])
                            {
                                switchOn[k]++;
                            }
                        }
                    }
                }
            }

            for(var n = 0; n < switchOn.Length; n++)
            {
                    if(switchOn[n] % 2 != int.Parse(checkRemaind[n]))
                    {
                        flag = false;
                    }
            }

            for(var m = 0; m < switchOn.Length; m++)
            {
                switchOn[m] = 0;
            }

            if (flag)
            {
                ans++;
            }
            flag = true;

        }

        Console.WriteLine(ans);
    }
}