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;

namespace CSharpSample01
{
    
    class Program
    {

        static List<string> button_combi;

        static void Create_button_combi(string str, int cnt, int state, int N)
        {
            string s = "";

            if (cnt != 0)
            {
                s = str + state.ToString();             
            }

            if (cnt >= N)
            {
                button_combi.Add(s);
                return;
            }
            
            Create_button_combi(s, cnt + 1, 1, N);
            
            Create_button_combi(s, cnt + 1, 0, N);

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

            button_combi = new List<string>();
            List<int>[] bulb = new List<int>[M];
            int[] bulb_val = new int[M];

            Create_button_combi("", 0, 0, N);

            for(int i = 0; i < M; i++)
            {
                int[] a = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();

                bulb[i] = new List<int>();

                for (int j = 1; j < a.Count(); j++)
                {
                    bulb[i].Add(a[j] - 1);
                }
            }

            bulb_val = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();

            int ans = 0;

            foreach(var bt_str in button_combi)
            {
                int cnt_bulb_on = 0;

                // i番目のバルブ
                for(int i = 0; i < M; i++)
                {

                    int cnt_bt_on = 0;

                    for(int j = 0; j < bulb[i].Count(); j++)
                    {
                        int bt_num = bulb[i][j];
                        
                        if(bt_str[bt_num] == '1')
                        {
                            cnt_bt_on++;
                        }                  

                    }
                    
                    if(cnt_bt_on % 2 == bulb_val[i])
                    {
                        cnt_bulb_on++;
                    }

                }

                
                if(cnt_bulb_on == M)
                {
                    ans++;
                }
            }



            Console.WriteLine(ans);
            
        }
    }
}