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.Linq;
using System.Collections.Generic;

class Program
{
    public static int ReadInt()
    {
        return int.Parse(Console.ReadLine());
    }

    public static int[] ReadInts()
    {
        return Console.ReadLine().Split().Select(int.Parse).ToArray();
    }

    public static void Main()
    {
        var input = ReadInts();
        var N = input[0];
        var M = input[1];
        var s = new List<int>[M];
        for (int i = 0; i < M; i++)
        {
            var input2 = ReadInts();
            var k = input2[0];
            s[i] = new List<int>();
            for (int j = 0; j < k; j++)
            {
                s[i].Add(input2[j + 1] - 1);
            }
        }
        var p = ReadInts();
        var count = 0;
        for (int i = 0; i < Math.Pow(2, N); i++)
        {
            var b = new bool[N];
            for (int j = 0; j < N; j++)
            {
                b[j] = (i & (1 << j)) != 0;
            }
            for (int j = 0; j < M; j++)
            {
                var c = 0;
                foreach (var item in s[j])
                {
                    if (b[item])
                    {
                        c++;
                    }
                }
                if (c % 2 != p[j])
                {
                    count--;
                    break;
                }
            }
            count++;
        }
        Console.WriteLine(count);
    }
}