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.IO;
using System.Linq;

namespace ABC128C
{
    class Program
    {
        int N;
        int M;
        int[,] sw;
        int[] target;
        int count = 0;

        static void Main(string[] args)
        {
            if (args.Length > 0)
            {
                Console.SetIn(new StreamReader(args[0]));
            }

            new Program().Solve();
        }

        void Solve()
        {
            var temp = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
            N = temp[0]; // スイッチの数
            M = temp[1]; // 電球の数
            sw = new int[M, N];

            for (int i = 0; i < M; i++)
            {
                temp = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
                var cnt = temp[0];

                for (int j = 1; j <= cnt; j++)
                {
                    sw[i, temp[j] - 1] = 1;
                }
            }

            target = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
            var li = new int[N];
            Search(0, li);
            Console.WriteLine(count);
        }

        void Search(int no, int[] li)
        {
            if (no == N)
            {
                // for (int i = 0; i < N; i++)
                // {
                //     Console.Write(li[i]);
                // }
                // Console.WriteLine();

                var fg = true;

                for (int i = 0; i < M; i++)
                {
                    var t = 0;

                    for (int j = 0; j < N; j++)
                    {
                        if (li[j] == 1 && sw[i, j] == 1)
                        {
                            t = (t + 1) % 2;
                            // Console.WriteLine("check: " + i + ":" + j + ":" + t);
                        }
                    }
                    // Console.WriteLine("t: " + t + ":" + target[i]);

                    if (t != target[i])
                    {
                        fg = false;
                        break;
                    }
                }

                if (fg)
                {
                    // Console.WriteLine("ok");
                    count++;
                }
            }
            else
            {
                li[no] = 0;
                Search(no + 1, li);

                li[no] = 1;
                Search(no + 1, li);
            }
        }
    }
}