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>
using namespace std;
using i64 = long long;
using u64 = unsigned long long;
#define REP(i, n) for (int i = 0; (i64)(i) < (i64)(n); ++i)

#ifdef ENABLE_DEBUG
template <typename T>
void debug(T value) {
  cerr << value;
}
template <typename T, typename... Ts>
void debug(T value, Ts... args) {
  cerr << value << ", ";
  debug(args...);
}
#define DEBUG(...)                              \
  do {                                          \
    cerr << " \033[33m (L" << __LINE__ << ") "; \
    cerr << #__VA_ARGS__ << ":\033[0m ";        \
    debug(__VA_ARGS__);                         \
    cerr << endl;                               \
  } while (0)
#else
#define debug(...)
#define DEBUG(...)
#endif

int main() {
  ios::sync_with_stdio(false);
  cin.tie(nullptr);
  int n, m;
  cin >> n >> m;
  vector<vector<int>> conn(n);
  REP(i, m) {
    int k;
    cin >> k;
    REP(j, k) {
      int s;
      cin >> s;
      s--;
      conn[s].push_back(i);
    }
  }
  vector<int> p(m);
  REP(i, m) { cin >> p[i]; }

  i64 ans = 0;
  const int L = 1LL << n;
  REP(state, L) {
    vector<int> lamp = p;
    REP(i, n) {
      if (state & (1 << i)) {
        for (int k : conn[i]) {
          lamp[k] ^= 1;
        }
      }
    }
    int num_on = 0;
    REP(i, m) {
      if (lamp[i] == 0) ++num_on;
    }
    if (num_on == m) {
      ++ans;
    }
  }
  cout << ans << endl;
}