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{
  static void Main(){
    string[] NM = Console.ReadLine().Split(' ');
    int N = int.Parse(NM[0]);
    int M = int.Parse(NM[1]);
    
    // スイッチの付け方の全組み合わせ (0 ~ 2^N-1)
    var list = new List<int>();
    for (int i = 0; i < Math.Pow(2, N); i++){
      list.Add(i);
    }
    var tmplist = list.ToList();
    
    int[,] s = new int[M+1, N+1];
    for (int i = 0; i < M; i++){
      string[] input = Console.ReadLine().Split(' ');
      int k = int.Parse(input[0]);
      for (int j = 0; j < k; j++){
        s[i, j] = int.Parse(input[j + 1]);
      }
    }
    
    string[] p = Console.ReadLine().Split(' ');
    
    // 全探索
    for (int i = 0; i < M; i++){
      int mod = int.Parse(p[i]);
      list = tmplist.ToList();
      foreach (int L in list){
        string bin = Convert.ToString(L, 2);
        int binl = bin.Length;
        for (int b = 0; b < N - binl; b++){ // bin の長さを N に合わせる
          bin = "0" + bin;
        }
        
        int cnt = 0;
        
        for (int j = 0; j < N; j++){
          if (s[i, j] == 0)
            break;
          else if (bin.Substring(s[i, j] - 1, 1) == "1")
            cnt++;
        }
        if (cnt % 2 != mod)
          tmplist.Remove(L);
      }
    }
    Console.WriteLine(tmplist.Count);
  }
}