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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#include <set>
#include <queue>

using namespace std;

#define rep(i, n) for (int i = 0; i < (int)(n); i++) 
#define reps(i, s, n) for (int i = s; i < (int)(n); i++) 
#define ll long long
#define ld long double
#define repl(i, n) for (ll i = 0; i < (ll)(n); i++) 
#define repsl(i, s, n) for (ll i = s; i < (ll)(n); i++)  
#define outdeb(fir, sec, thr) cout << fir << ":" << sec << ":" << thr << endl

#define lpair pair<ll, ll>

int main()
{
  ll n,m;
  cin >> n >> m;
  vector<ll> rits;
  vector<ll> outs;
  repl(i, m)
  {
    ll k;
    cin >> k;
    ll d = 0;
    repl(j, k)
    {
      ll tmp;
      cin >> tmp;
      d += 1 << (tmp-1);
    }
    rits.push_back(d);
  }
  
  repl(i, m)
  {
    ll t;
    cin >> t;
    outs.push_back(t);
  }

  ll ans = 0;
  repl(i, pow(2, n))
  {
    bool isError = false;
    repl(j, m)
    {
      ll tmp = rits[j] & i;
      if(__builtin_popcountll(tmp) % 2 != outs[j] )
      {
        isError = true;
        break;
      }
    }
    if(!isError) ans++;
  }

  cout << ans << endl;

  return 0;
}