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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
#include <iomanip>
using namespace std;

int main(){
  int N,M; cin >> N >> M;
  vector<vector<int>> k_s(M,vector<int>(N+1));
  vector<int> p(M,0);
  for(int i = 0; i < M; i++){
      int k;
      cin >> k ;
      k_s.at(i).at(0) = k;
      for(int j = 1; j < k + 1; j++){
          cin >> k_s.at(i).at(j);
      }
  }
  
  /*for(int i = 0; i < M; i++){
      for(int j = 0; j < N+1; j++){
          cout << k_s.at(i).at(j) << ' ';
      }
      cout << endl;
  }*/
  int ON = 0;
  for(int i = 0; i < M; i++){
      cin >> p.at(i);
  }
  
  for(int bit = 0; bit < (1 << N); bit++){
      
      vector<bool> S(N,false);
      for(int i = 0; i < N; i++){
          if(bit & (1 << i)){
              S.at(i) = true;
          }
      }
      /*for(int i = 0; i < N; i++){
          if(S.at(i) == true) cout << 1;
          else cout << 0;
      }
      cout << endl;*/
      
      bool allOK = true;
     
      for(int i = 0; i < M; i++){
          int n_on = 0;
           for(int j = 1; j < k_s.at(i).at(0) + 1; j++){
              bool a = S.at(k_s.at(i).at(j)-1);
              if(a) n_on++;
          }
          if(n_on % 2 != p.at(i)) allOK = false;
      }
      if(allOK) ON++;
  }
  cout << ON << endl;
}