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 (1.42.0) (judge verdict: Accepted)

#![allow(non_snake_case)]
#![allow(unused_imports)]

use proconio::{input, fastout};
use proconio::marker::*;
use whiteread::parse_line;
use std::collections::*;
use num::*;
use num_traits::*;
use superslice::*;
use std::ops::*;

use itertools::Itertools;
use itertools_num::ItertoolsNum;

#[fastout]
fn main() {
    let (n, m): (usize, usize) = parse_line().unwrap();
    let mut s_vec: Vec<Vec<usize>> = Vec::new(); 
    for _ in 0..m {
        let v: Vec<usize> = parse_line().unwrap();
        s_vec.push(v[1..].to_vec());
    }

    let p_vec: Vec<usize> = parse_line().unwrap();
    
    let mut ans = 0;

    for bit in 0..1<<n {
        let mut vec: Vec<usize> = Vec::new();
        for i in 0..n {
            if bit & (1<<i) != 0 {
                vec.push(i+1);
            }
        }

        // if vec.len() == 0 {
        //     continue;
        // }

        let mut flag = true;
        for i in 0..m {
            let set: HashSet<usize> = vec.clone().into_iter().collect();
            let s_set: HashSet<usize> = s_vec[i].clone().into_iter().collect();
            let intersect: Vec<&usize> = s_set.intersection(&set).collect::<Vec<&usize>>();
            if intersect.len() % 2 != p_vec[i] {
                flag = false;
                break;
            }
        }
        if flag {
            ans += 1;
        }
    }

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