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 rep(i,j,n) for (ll i = j; i < (n); i++)
#define CLR(mat,f) memset(mat, f, sizeof(mat))
#define IN(a, b, x) (a<=x&&x<b)
#define out(ans) cout << ans << endl
typedef pair<ll,ll>P;
const ll mod=1e9+7;
const int INF = 1000000;
const double PI=3.14159265359;

int main(){

  ll n,m;cin>>n>>m;
  vector<vector<ll>>v(m);
  rep(i,0,m){
    ll a;cin>>a;
    rep(j,0,a){
      ll b;cin>>b;
      v[i].push_back(--b);
    }
  }
  bool p[m];rep(i,0,m)cin>>p[i];


  ll ans=0;
  rep(bit,0,(1<<n)){
    bool e=true;
    rep(i,0,m){
      ll count=0;
      // out(i<<":"<<v[i].size());
      rep(j,0,v[i].size()){
        // if((bit>>v[i][j])&1)count++;
        if(bit&(1<<v[i][j]))count++;
      }
      if(count%2!=p[i])e=false;
    }
    if(e)ans++;
  }

  out(ans);

  return 0;
}