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)

struct Bulb {
    p: usize,
    switches: Vec<usize>,
}

fn dfs(switches: &mut Vec<usize>, bulbs: &Vec<Bulb>, n: usize, now_n: usize) -> usize {
    let mut sum = 0usize;

    for i in 0..2 {
        switches[now_n] = i;

        if now_n == n-1 {
            let mut light_all = true;
            for bulb in bulbs {
                let mut switch_sum = 0;
                for switch in &bulb.switches {
                    switch_sum += switches[*switch - 1];
                }
                if switch_sum % 2 != bulb.p {
                    light_all = false;
                    break;
                }
            }

            if light_all {
                sum += 1;
            }
        }
        else {
            sum += dfs(switches, bulbs, n, now_n+1);
        }
    }

    sum
}

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_line(&mut buf).unwrap();
    let mut iter = buf.split_whitespace();
    let n: usize = iter.next().unwrap().parse().unwrap();
    let m: usize = iter.next().unwrap().parse().unwrap();


    let mut bulbs = Vec::new();
    for _i in 0..m {
        let mut buf = String::new();
        std::io::stdin().read_line(&mut buf).unwrap();
        let mut iter = buf.split_whitespace();
        let _k: usize = iter.next().unwrap().parse().unwrap();
        let switches = iter.map(|x| x.parse::<usize>().unwrap()).collect::<Vec<_>>();
        bulbs.push(Bulb{p: 0, switches: switches});
    }

    let mut buf = String::new();
    std::io::stdin().read_line(&mut buf).unwrap();
    let mut iter = buf.split_whitespace();
    for i in 0..m {
        bulbs[i].p = iter.next().unwrap().parse().unwrap();
    }

    let mut switches = vec![0; n];
    let ans = dfs(&mut switches, &bulbs, n, 0);

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