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 (n, m) = {
        let nm = read_vec::<usize>();
        (nm[0], nm[1])
    };
    let mut lights: Vec<Vec<usize>> = Vec::new();
    for i in 0..m {
        let ks = read_vec::<usize>();
        let k = ks[0];
        let mut v = vec![0; k];
        for j in 0..k {
            v[j] = ks[j+1] - 1;
        }
        lights.push(v);
    }
    let p = read_vec::<usize>();

    // switchをbit全探索
    let mut ret = 0;
    for i in 0..(1 << n) {
        let mut switches = vec![0; n];
        for j in 0..n {
            let bit = 1 << j;
            if (i & bit) == 0 { continue; }
            switches[j] = 1; // switch on
        }

        let mut on_lights = 0;
        for (i_light, light) in lights.iter().enumerate() {
            let mut cnt = 0;
            for j in light {
                if switches[*j] == 1 { cnt += 1; }
            }
            if cnt % 2 == p[i_light] { on_lights += 1; }
        }

        if on_lights == m { ret += 1; }
    }

    println!("{}", ret);
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().split_whitespace().map(|e| e.parse().ok().unwrap()).collect()
}