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;
typedef pair<ll, ll> P;
typedef pair<pair<int, int>, int> PP;
const ll INF = 1LL << 60;
const ll MOD = 1000000007;

int GCD(int a, int b) { return b ? GCD(b, a%b) : a; }
//------------------------------------------------------------------------------------------------------------------------------

int N, M;
bool on[20];
vector<int> p(20);
vector<int> V[20];

int func(int s) {
    int ret = 0;
    if (s == N) {
        bool flag = true;
        for (int i = 0; i < M; i++) {
            int sz = V[i].size();
            int cnt = 0;
            for (int j = 0; j < sz; j++) {
                if (on[V[i][j]]) cnt++;
            }
            if (cnt % 2 != p[i]) flag = false;
        }
        if (flag) return 1;
        return 0;
    }
    on[s] = true;
    ret += func(s + 1);
    on[s] = false;
    ret += func(s + 1);
    return ret;
}
int main(void) { 
    cin >> N >> M; 
    for(int i = 0; i < M; i++) {
        int k;
        cin >> k;
        for(int j = 0; j < k; j++) {
            int s;
            cin >> s;
            s--;
            V[i].push_back(s);
        }
    }
    for(int i = 0; i < M; i++) {
        int t;
        cin >> t;
        p[i] = t;
    }
    cout << func(0) << endl;
}