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;
using System.Collections.Generic;

namespace ABC128
{
    class Program
    {
        static int Search(int[] ss, int num, int M, List<int[]> s, int[] p)
        {
            if (num == ss.Length)
            {
                int res = 0;
                bool flag1 = true;
                for (int i = 0; i < M; i++)
                {
                    int cnt1 = 0;
                    for (int j = 0; j < s[i].Length; j++)
                    {
                        cnt1 += ss[s[i][j] - 1];
                    }
                    if (cnt1 % 2 != p[i]) flag1 = false;
                }
                if (flag1) res++;
                return res;
            }
            int[] ss1 = new int[ss.Length];
            int[] ss2 = new int[ss.Length];
            Array.Copy(ss, ss1, ss.Length);
            Array.Copy(ss, ss2, ss.Length);
            ss1[num] = 0;
            ss2[num] = 1;
            return Search(ss1, num + 1, M, s, p) + Search(ss2, num + 1, M, s, p);
        }
        static void Main(string[] args)
        {
            string[] S = Console.ReadLine().Split(' ');
            int N = int.Parse(S[0]);
            int M = int.Parse(S[1]);

            int[] k = new int[M];
            List<int[]> s = new List<int[]>();

            for (int i = 0; i < M; i++)
            {
                S = Console.ReadLine().Split(' ');
                k[i] = int.Parse(S[0]);
                int[] ss = new int[k[i]];
                for (int j = 0; j < k[i]; j++)
                {
                    ss[j] = int.Parse(S[j + 1]);
                }
                s.Add(ss);
            }
            int[] p = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();

            int res = 0;
            res = Search(new int[N], 0, M, s, p);
            
            Console.WriteLine(res);
        }
    }
}