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>
using namespace std;
 
int main() {
  int n,m;
  cin >> n >> m;
  vector<int> k(m);
  vector<vector<int>> s(m,vector<int>(10));
  int tmp;
  int i,j;
  for(i=0;i<m;i++){
    cin >> k[i];
    for(j=0;j<k[i];j++){
      cin >> tmp;
      s[i][tmp-1] = 1;
    }
  }
  vector<int> p(m);
  for(i=0;i<m;i++){
    cin >> p[i];
  }
  int mp[10];
  int onoff = 0,ans = 0;
  int a,b,c,d,e,f,g,h,l;
  vector<int> z(10);
  for(i=0;i<n;i++){
    z[i] = 1;
  }
  for(a=0;a<z[0] +1;a++){
    for(b=0;b<z[1] +1;b++){
      for(c=0;c<z[2] +1;c++){
        for(d=0;d<z[3] +1;d++){
          for(e=0;e<z[4] +1;e++){
            for(f=0;f<z[5] +1;f++){
              for(g=0;g<z[6] +1;g++){ 
                for(h=0;h<z[7] +1;h++){    
                  for(i=0;i<z[8] +1;i++){
                    for(j=0;j<z[9] +1;j++){
                      tmp = 0;
                      for(l=0;l < m;l++){
                        onoff = a * s[l][0] + b * s[l][1] + c * s[l][2] + d * s[l][3] + e * s[l][4] + f * s[l][5] + g * s[l][6] + h * s[l][7] + i * s[l][8] + j * s[l][9];

                        if(onoff % 2 == p[l]){
                          tmp++;
                        }
                      }
                      ans += tmp/m;
                    }
                  }
                }
              }
            }
          }
        }
      }
    }
  }
  cout << ans << endl;
}