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>


using namespace std;

int N, M;
int switches[10];
int ans = 0;
vector<int> states;
vector<vector<int>> lamps;


//全探索
void DFS(int k) {
	if (k < 0) {
		//すべての電球が点灯するか
		int add = 1;
		for (int i = 0; i < M; i++)
		{
			int count = 0;
			for (auto j : lamps[i]) {
				count += switches[j - 1];
			}
			if (count % 2 != states[i]) {
				add = 0;
				break;
			}
		}
		ans += add;
		return;
	}
	else {
		switches[k] = 0;
		DFS(k - 1);
		switches[k] = 1;
		DFS(k - 1);
		return;
	}
}

int main() {

	cin >> N >> M;

	lamps.resize(M);

	for (int i = 0; i < M; i++)
	{
		int k;
		cin >> k;
		for (int j = 0; j < k; j++)
		{
			int s;
			cin >> s;
			lamps[i].push_back(s);
		}
	}

	for (int i = 0; i < M; i++) {
		int p;
		cin >> p;
		states.push_back(p);
	}

	DFS(N - 1);

	cout << ans << endl;

	return 0;
}