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;

int main()
{
	int i, j, n, m, x, ans=0;
	cin >> n >> m;
	int a[n], p[m], flag[m], k[m], count;
	int t1=pow(2,n), t2, t3;
	vector< vector<int> > v(m);

	for(i=0; i<m; i+=1)
	{
		cin >> k[i];
		for(j=0; j<k[i]; j+=1)
		{
			cin >> x;
			v[i].push_back(x-1);
		}
	}
	for(i=0; i<m; i+=1)
		cin >> p[i];

	for(i=0; i<t1; i+=1)
	{
		memset(a, 0, sizeof(a));
		memset(flag, 0, sizeof(flag));
		x=i;
		t2=0;
		while(x!=0)
		{
			a[t2]=x%2;
			x/=2;
			t2+=1;
		}
		for(j=0; j<m; j+=1)
		{
			count=0;
			for(t3=0; t3<k[j]; t3+=1)
				if(a[v[j][t3]]==1)
					count+=1;
			if(count%2==p[j])
				flag[j]=1;
		}
		for(j=0; j<m; j+=1)
		{
			if(flag[j]==0)
			{
				ans-=1;
				break;
			}
		}
		ans+=1;
	}
	cout << ans << endl;
}