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 Ruby (2.3.3) (judge verdict: Runtime Error)

#include <bits/stdc++.h>

using namespace std;
typedef long long ll;

#define INF (long long)(1e9+7)
#define REP(i,n) for(ll i = 0; i < n; i++)
#define REP1(i,n) for(ll i = 1; i <= n; i++)

int k[20], s[20][20], p[20];

int main(void)
{
    int n, m;
    cin >> n >> m;
    REP(i, m) {
        cin >> k[i];
        REP(j, k[i]) {
            cin >> s[i][j];
            s[i][j] --;
        }
    }
    REP(i, m) cin >> p[i];
    ll cnt = 0;
    for(int sw = 0; sw < (1<<n); sw++) {
//        cout << "sw:" << sw << endl;
        bool all=true;
        REP(i, m) {
            int on = 0;
            REP(j, k[i]) {
                if(sw & (1 << s[i][j])) on ++;
            }
            if(on % 2 != p[i]) all = false;
        }
        if(all) cnt ++; 
//        cout << "cnt:" << cnt << endl;
    }
    cout << cnt << endl;
}