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

namespace debug
{
    class ProgramD
    {
        static void Main(string[] args)
        {
            string[] t = Console.ReadLine().Split(' ');
            int n = int.Parse(t[0]);
            int m = int.Parse(t[1]);

            //スイッチがそれぞれオンオフを考える
            int[,] sw = new int[m,n];
            int[] power = new int[m];
            int[] count = new int[m];
            int ans = 0;

            //スイッチを押すと反応する電球を記憶
            for (int i = 0; i < m; i++)
            {
                string[] s = Console.ReadLine().Split(' ');
                for(int j = 0; j < int.Parse(s[0]);j++)
                    sw[i, int.Parse(s[j + 1]) - 1] = 1;
                
            }

            //最後のオンオフの条件も記録
            string[] u = Console.ReadLine().Split(' ');
            for (int i = 0; i < m; i++)
                power[i] = int.Parse(u[i]);


            //ビット全探索
            for (int i = 0; i < (1 << n);i++)
            {
                //まずはオンオフでオンしている数を数える
                for(int j = 0; j < n;j++)
                {
                    if(((i >> j) & 1) == 1)
                    {
                        for(int k = 0;k < m;k++)
                        {
                            if (sw[k, j] == 1)
                                count[k]++;
                        }

                    }
                }

                //もし一つでもオン状態がなければアウト
                for(int j = 0;j < m;j++)
                {
                    if (count[j] % 2 != power[j])
                        break;

                    if (j == m - 1)
                        ans++;

                }

                for(int j = 0; j < m;j++)
                    count[j] = 0;
            }

            Console.WriteLine(ans);
            return;
        }
    }

}