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;

public static class Program
{
    public static void Main(string[] args)
    {
        var nm = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();
        var vals = new List<int[]>();
        for (int i = 0; i < nm[1]; i++)
        {
            var val = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();
            vals.Add(val.Where((value, index) => index != 0).ToArray());
        }
        var p = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();
        var cnt = 0;
        for (int i = 0; i < Math.Pow(2, nm[0]); i++)
        {
            var ary = new bool[nm[0]];
            for (int j = 0; j < nm[0]; j++)
            {
                var y = (i >> j) & 1;
                ary[j] = y == 1;
            }
            var flg = vals.Select((v, x) => Check(ary, v, p[x])).ToArray();
            if (flg.Count(x => x) == nm[1])
            {
                cnt++;
            }
        }
        Console.WriteLine(cnt);
    }

    public static bool Check(bool[] ary, int[] val, int p)
    {
        var cnt = val.Count(x => ary[x - 1] == true);
        return cnt % 2 == p ? true : false;
    }
}