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: Wrong Answer)

#![allow(dead_code)]
#![allow(non_snake_case)]
// Original macro is in https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
// Thanks @tanakh :bow:
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = || -> String {
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}
 
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};
 
    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
    
    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}
 
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };
 
    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, [$t:tt]) => {
        {
            let len = read_value!($next, usize);
            (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
        }
    };
 
    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };
 
    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };
 
    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

// for adhoc read (ex. https://atcoder.jp/contests/abc112/tasks/abc112_a)
fn read_line() -> String {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).unwrap();
    s.trim().to_string()
}

#[derive(Debug)]
struct Matrix<T=f32> {
    row: usize,
    col: usize,
    matrix: Vec<Vec<T>>,
}
trait GaussJordan {
    fn rank(&mut self) -> usize;
}
impl GaussJordan for Matrix {
    fn rank(&mut self) -> usize {
        let mut rank = 0;

        for col in 0..self.col {
            let mut pivot = 1000000;
            let mut ma = 1e-10;
            for row in rank..self.row {
                if self.matrix[row][col].abs() > ma {
                    ma = self.matrix[row][col].abs();
                    pivot = row;
                }
            }
            if pivot == 1000000 {
                continue;
            }
            self.matrix.swap(pivot, rank);

            let div = self.matrix[rank][col];
            for col2 in 0..self.col { self.matrix[rank][col2] /= div; }
            for row in 0..self.row {
                if row != rank && self.matrix[row][col].abs() > 1e-10 {
                    let fac = self.matrix[row][col];
                    for col2 in 0..self.col {
                        self.matrix[row][col2] -= self.matrix[rank][col2] * fac;
                    }
                }
            }
            rank += 1;
            // println!("---------------------");
            // println!("col {}", col);
            // println!("pivot {}, rank {}", pivot, rank);
            // println!("  swapped matrix {:?}", self.matrix);
        }
        return rank;
    }
}
#[test]
fn matrix_rank_test() {
    let mat = vec![vec![4.0, 2.0, 3.0, 4.0],
                   vec![1.0, 2.0, 3.0, 4.0],
                   vec![0.0, 0.0, 0.0, 0.0]];
    let mut m = Matrix::<f32>{ row: mat.len(), col: mat[0].len() , matrix: mat };
    assert_eq!(2, m.rank());
}
#[test]
fn f2_rank_test() {
    let mat = vec![vec![1.0, 1.0, 1.0, 1.0],
                   vec![0.0, 1.0, 1.0, 1.0],
                   vec![0.0, 0.0, 1.0, 0.0]];
    let mut m = Matrix::<f32>{ row: mat.len(), col: mat[0].len() , matrix: mat };
    assert_eq!(3, m.rank());
}

// https://practice.contest.atcoder.jp/tasks/practice_1
fn main() {
    input!{
        N: usize, 
        M: usize,
        denkyus: [[usize]; M],
        _p: [usize; M]
    }
    let mut mat = vec![vec![0.0; N]; M];
    for (i, arr) in denkyus.iter().enumerate() {
        for switch in arr {
            mat[i][switch-1] = 1.0;
        }
    }
    // println!("mat {:?}", mat);
    let rank = Matrix::<f32>{ row: M, col: N, matrix: mat }.rank();
    println!("{}", 1 << N-rank );
}