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)

use std::io::Read;

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();

    let answer = solve(&buf);

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

fn solve(input: &str) -> String {
    let mut iterator = input.split_whitespace();

    let n: usize = iterator.next().unwrap().parse().unwrap();
    let m: usize = iterator.next().unwrap().parse().unwrap();

    let mut lumps: Vec<Vec<usize>> = Vec::with_capacity(m);
    for _i in 0..m {
        let mut vec: Vec<usize> = Vec::new();
        let k: usize = iterator.next().unwrap().parse().unwrap();
        for _j in 0..k {
            vec.push(iterator.next().unwrap().parse::<usize>().unwrap() - 1);
        }
        lumps.push(vec);
    }

    let mut p: Vec<usize> = Vec::with_capacity(m);
    for _i in 0..m {
        p.push(iterator.next().unwrap().parse().unwrap());
    }

    let mut ans = 0;
    for i in 0..(2_usize.pow(n as u32)) {
        let pattern = make_pattern(i, n);
        let mut is_ok = true;
        for (j, lump) in lumps.iter().enumerate() {
            let mut count = 0;
            for &l in lump {
                if pattern[l] {
                    count += 1;
                }
            }
            if count % 2 != p[j] {
                is_ok = false;
                break;
            }
        }
        if is_ok {
            ans += 1;
        }
    }

    return ans.to_string();
}

fn make_pattern(i: usize, len: usize) -> Vec<bool> {
    let mut n = i;
    let mut result = Vec::with_capacity(len);
    while n > 0 {
        if n % 2 == 1 {
            result.push(true);
        } else {
            result.push(false);
        }
        n /= 2;
    }
    while result.len() < len {
        result.push(false);
    }
    result
}