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.Text;
using System.Numerics;
using System.IO;

namespace SolutionCS
{

    class Program
    {
        static void Main(string[] args)
        {
            string[] s = Console.ReadLine().Split(' ');
            int N = int.Parse(s[0]);
            int M = int.Parse(s[1]);

            int maxSW = (int)Math.Pow(2, N);

            int[] count = new int[M];
            int[] mask = new int[M];

            for (int i = 0; i < M; i++)
            {
                s = Console.ReadLine().Split(' ');

                int K = int.Parse(s[0]);

                for (int j = 0; j < K; j++)
                {
                    mask[i] |= (1 << (int.Parse(s[j + 1]) -1));
                }
            }

            s = Console.ReadLine().Split(' ');
            for (int i = 0; i < M; i++)
            {
                count[i] = int.Parse(s[i]);
            }

            int ans = 0;
            for (int i = 0; i < maxSW; i++)
            {
                bool isOn = true;
                for (int j = 0; j < M; j++)
                {
                    int r = i & mask[j];
                    if(count[j] != bitCount(r) % 2)
                    {
                        isOn = false;
                        break;
                    }
                }

                if (isOn)
                {
                    ans++;
                }
            }

            Console.WriteLine(ans);
        }


        static int[] bitC = new int[]
        {
            0,1,1,2,
            1,2,2,3,
            1,2,2,3,
            2,3,3,4
        };

        static int bitCount(int i)
        {
            int c = 0;
            while(i>0)
            {
                c += bitC[i & 0x0f];
                i >>= 8;
            }

            return c;
        }
    }
}