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 <iostream>
#include <vector>
#include <cmath>

using namespace std;

int main() {
	int n, m; cin >> n >> m;
	vector<vector<bool> > s(m, vector<bool>(n + 1, false));
	vector<int> p(m);
	for (int i = 0; i < m; ++i) {
		int k; cin >> k;
		for (int j = 0; j < k; ++j) {
			int t; cin >> t;
			s[i][t] = true;
		}
	}
	for (int i = 0; i < m; ++i) cin >> p[i];

	int ans = 0;
	vector<int> swio (n);
	for (int i = 0; i < (int)pow(2, n); ++i) {
		int tmp = i;
		for (int j = 0; j < n; ++j) {
			swio[j] = tmp % 2;
			tmp /= 2;
		}
		bool flag = true;
		for (int j = 0; j < m; ++j) {
			int cnt = 0;
			for (int k = 1; k <= n; ++k) {
				if (swio[k - 1] && s[j][k]) cnt++;
			}
			if (cnt % 2 != p[j]) {
				flag = false;
				break;
			}
		}
		if (flag) ans++;
	}
	cout << ans << endl;


	return 0;
}