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>
#include <vector>
#include <string>

using namespace std;

vector<string> split(const string &s, char delim) {
  vector<string> elems;
  string item;
  for (char ch: s) {
    if (ch == delim) {
      if (!item.empty()) {
        elems.push_back(item);
      }
      item.clear();
    }
    else {
      item += ch;
    }
  }
  if (!item.empty()) {
    elems.push_back(item);
  }
  return elems;
}

string to_str_with_zero(int i, int w) {
  ostringstream sout;
  sout << std::setfill('0') << std::setw(w) << i;
  string s = sout.str();
  return s;
}

int main() {
  std::cout << std::setprecision(9);
  int n, m;
  cin >> n >> m;

  vector<bitset<10>> stols(m);
  for (int i=0; i<m; i++) {
    int k;
    cin >> k;
    for (int j=0; j<k; j++) {
      int s;
      cin >> s;
      stols.at(i).set(s - 1, 1);
    };
  }

  vector<int> p(m);
  for(int i=0; i<m; i++) {
    cin >> p.at(i);
  }

  int ans = 0;
  for (int tmp=0; tmp < (1 << 10); tmp++) {
    if (tmp >= pow(2, n)) {
      break;
    }
    bitset<10> lightset(tmp);
    
    bool all_on = true;
    for (int i=0; i<m; i++) {
      bitset<10> on = lightset & stols.at(i);
      if (on.count() % 2 != p.at(i)) {
        all_on = false;
      }
    }
    if (all_on) {
      ans++;
    }
  }

  cout << ans << endl;
}