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;
use std::str::FromStr;

fn input_int_tuple() -> (i32, i32) {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    let vec: Vec<i32> = line.split_whitespace().map(|s| s.parse().unwrap()).collect();
    (vec[0], vec[1])
}

fn input_i32_vec() -> Vec<i32> {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    line.split_whitespace().map(|s| s.parse().unwrap()).collect()
}

fn dfs(i: &mut i32, switches_condition: &Vec<Vec<i32>>, p_vec: &Vec<i32>, switches: &mut Vec<bool>, count: &mut i32) {
    if *i == switches.len() as i32 {
        // println!("{:?}", switches);
        let mut flag = true;
        for light_idx in 0..switches_condition.len() {
            let mut switch_count = 0;
            for target in &switches_condition[light_idx] {
                if switches[(target - 1) as usize] == true {
                    switch_count += 1;
                }
            }
            if switch_count % 2 != p_vec[light_idx] {
                flag = false;
                break;
            }
        }
        if flag {
            *count += 1;
        }
        return;
    }
    switches[*i as usize] = true;
    dfs(&mut (*i + 1), switches_condition, p_vec, switches, count);
    switches[*i as usize] = false;
    dfs(&mut (*i + 1), switches_condition, p_vec, switches, count);
}

fn main() {
    let (n, m) = input_int_tuple();
    let mut switches_condition: Vec<Vec<i32>> = Vec::new();
    for i in 0..m {
        let mut condition = input_i32_vec();
        condition.remove(0);
        switches_condition.push(condition);
    }
    // println!("{:?}", switches_condition);
    let p_vec = input_i32_vec();
    let mut count = 0;
    dfs(&mut 0, &switches_condition, &p_vec, &mut vec![false; n as usize], &mut count);
    println!("{}", count);
}