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;
typedef long long ll;
#define MOD 1000000007

int main(){
  ll n, m;
  cin >> n >> m;
  vector<vector<int>> s(m+1);
  vector<int> k(m+1);
  for(int i=1; i<=m; i++){
    cin >> k[i];
    s[i].assign(n+1, 0);
    for(int j=1; j<=k[i]; j++){
      int tmp;
      cin >> tmp;
      s[i][tmp] = 1;
    }
  }
  vector<int> p(m+1);
  for(int i=1; i<=m; i++){
  	cin >> p[i];
  }
  
  int ans = 0;
  for(int bit=0; bit<(1<<n); ++bit){
    vector<int> pls(m+1, 0);
    for (int i=0; i<n; ++i) {
      if (bit & (1<<i)){
        for(int j=1; j<=m; j++){
          pls[j] += s[j][i+1];
        }
      }
    }
    int how = 0;
    for(int i=1; i<=m; i++){
      if(pls[i] % 2 == p[i]){
        how++;
      }
    }
    if(how == m){
      ans++;
    }
  }
  cout << ans << endl;
}