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 <algorithm>
#include<iostream>
#include<vector>
#include<map>
#include<set>
#include<string>

#include<stdlib.h>
#include<string.h>
#include<math.h>
#include<limits.h>




using namespace std;

struct switches{
  long k;
  vector<long> s;
  //long s;
};

int main(){
  long n,m,ii,jj,tmp;
  vector<switches> sw;
  vector<long>     light;
  cin >> n >> m;

  sw.resize(m);
  light.resize(m);

  for(ii=0;ii<m;ii++){
    cin >> sw[ii].k;
    sw[ii].s.resize(n);

    for(jj=0;jj<n;jj++){
      sw[ii].s[jj] = 0;
    }

    for(jj=0;jj<sw[ii].k;jj++){
      cin >> tmp;
      //cout << "tmp:" << tmp << endl;
      sw[ii].s[tmp-1] = 1;
    }

  }

  //cout << "sw end" << endl;
  for(ii=0;ii<m;ii++){
    //cout << "ii:" << ii << endl;
    cin >> light[ii];
  }

  for(ii=0;ii<m;ii++){
    for(jj=0;jj<n;jj++){
      //cout << "sws" << sw[ii].s[jj] << ":" ;
    }
    //cout << endl;
  }

  //cout << "end of input" << endl;
  vector<long> state;
  long kk,ll,swstate,res=0;
  vector<long> arr;
  state.resize(n);
  arr.resize(m);
  
  for(ii=0;ii<pow(2,n);ii++){
    tmp = ii;
    //cout << "tmp:"<< tmp << endl;;
    //スイッチノ状態を作成
    for(jj=0;jj<n;jj++){
      state[jj] = tmp % 2;
      tmp = tmp / 2;
      //cout << state[jj] << ":" ;
    }
    
    //cout << endl;

    for(ll=0;ll<m;ll++){
      arr[ll] = 0;
    }
    //スイッチの状態で全ての電灯がつくかを確認
    //for(jj=0;jj<n;jj++){
    swstate = 0;

    //for(kk=0;kk<n;kk++){
    //  if(state[kk] == 1){
    //	for(ll=0;ll<n;ll++){
    //	  if(sw[kk].s[ll] == 1){
    //	    arr[ll]++ ;
    //	  }
    //	}
    //	//cout << "jj:"  << jj << "kk:" << kk << endl;
    //  }
    //}

    for(kk=0;kk<m;kk++){
      for(ll=0;ll<n;ll++){
	if(state[ll] == 1 && sw[kk].s[ll] == 1){
	  arr[kk]++;
	}
      }
    }
    tmp = 0;
    
    for(kk=0;kk<m;kk++){
      if((light[kk] % 2) == (arr[kk] % 2)){
	tmp++;
      }
    }

    if(tmp == m){
      res++;
    }
    //for(ll=0;ll<m;ll++){
    //  cout << "arr:" << arr[ll] << endl;
    //}
    //cout << "state" << ii << endl;
  }
  cout << res << endl;
  return 0;
}