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 Rust (1.15.1) (judge verdict: Accepted)

fn main() {
    let stdin = std::io::stdin();
    let (n, m) = {
        let mut buf = String::new();
        stdin.read_line(&mut buf).unwrap();
        let mut iter = buf.split_whitespace().map(|x| x.parse::<usize>().unwrap());
        (iter.next().unwrap(), iter.next().unwrap())
    };
    let link = {
        let mut l = vec![vec![0; m]; n];
        for i in 0..m {
            let mut buf = String::new();
            stdin.read_line(&mut buf).unwrap();
            for s in buf
                .split_whitespace()
                .skip(1)
                .map(|x| x.parse::<usize>().unwrap())
            {
                l[s - 1][i] = 1;
            }
        }
        l
    };
    let p = {
        let mut buf = String::new();
        stdin.read_line(&mut buf).unwrap();
        buf.split_whitespace()
            .map(|x| x.parse::<usize>().unwrap())
            .collect::<Vec<usize>>()
    };
    let mut valid = 0;
    for switch in 0..(2u32).pow(n as u32) {
        let mut state = p.clone();
        for i in 0..n {
            if switch & (2u32).pow(i as u32) != 0 {
                for j in 0..m {
                    state[j] += link[i][j];
                }
            }
        }
        if state.iter().all(|x| x % 2 == 0) {
            valid += 1;
        }
        //println!("{:?}", state);
    }
    println!("{}", valid);
}