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: Wrong Answer)

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

namespace ABC128_C_Switches
{
	class Program
	{
		static void Main(string[] args)
		{
			string[] nm = Console.ReadLine().Split(' ');
			int n = int.Parse(nm[0]);
			int m = int.Parse(nm[1]);
			int[] k = new int[m];
			List<int[]> switcheList = new List<int[]>();
			for(int i = 0; i < m; i++)
			{
				int[] input = Console.ReadLine().Split(' ').Select(s => int.Parse(s)).ToArray();
				k[i] = input[0];
				int[] switches = new int[k[i]];
				for(int j = 0; j < k[i]; j++)
				{
					switches[j] = input[j + 1];
				}
				switcheList.Add(switches);
			}
			int[] p = Console.ReadLine().Split(' ').Select(s => int.Parse(s)).ToArray();

			int ans = 0;
			for(int i = 0; i < Math.Pow(2, n); i++)
			{
				int lightCnt = 0;
				char[] status = Convert.ToString(i, 2).PadLeft(n, '0').ToCharArray();
				for(int j = 0; j < m; j++)
				{
					int onCnt = 0;
					foreach (int x in switcheList[j])
					{
						if(status[x - 1] == '1')
						{
							onCnt++;
						}
					}
					if(onCnt % 2 == p[m - 1])
					{
						lightCnt++;
					}
				}
				if(lightCnt == m)
				{
					ans++;
				}
			}

			Console.WriteLine(ans);
		}
	}
}