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.Generic;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;

namespace kp
{
    public class Program
    {
        //static long[,] cmb;
        static int MAX_INT = 1000000007;
        static void Main(string[] args)
        {
            //int N = Int32.Parse(Console.ReadLine());
            int[] input = Console.ReadLine().Split().Select(p => int.Parse(p)).ToArray();
            int N = input[0];
            int M = input[1];

            // M行読み込む
            List<List<int>> switchState = new List<List<int>>();
            for(int i=0; i<M; i++)
            {
                input = Console.ReadLine().Split().Select(p => int.Parse(p)).ToArray();
                List<int> s = new List<int>();
                for(int j=0; j<input[0]; j++)
                {
                    s.Add(input[j + 1]);
                }
                switchState.Add(s);
            }
            // 電球が点灯する状態を読み込む
            input = Console.ReadLine().Split().Select(p => int.Parse(p)).ToArray();

            // 2^N通りのスイッチの状態で、電球が全てつくかチェックする
            int end = (int)Math.Pow((double)2, (double)(N));
            int ret = 0;
            for(int i=0; i<end; i++)
            {
                string bi = Convert.ToString(i, 2).PadLeft(N, '0');

                // 全ての電球が点灯するかチェックする
                bool allLit = true;
                for(int j=0; j<M; j++)
                {
                    List<int> s = switchState[j];
                    int litnum = 0;
                    for(int k=0; k<s.Count; k++)
                    {
                        if (bi[s[k] - 1] == '1')
                            litnum++;
                    }
                    if(litnum%2 != input[j])
                    {
                        allLit = false;
                        break;
                    }
                }
                if (allLit)
                    ret++;
            }
            Console.WriteLine(ret + "");
        }
    }
}