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)

fn main(){
  let nm: Vec<usize> = read_vec();
  let n = nm[0];
  let m = nm[1];

  let mut sv: Vec<Vec<usize>> = vec![Vec::new(); m];
  
  for i in 0 .. m {
    let tv: Vec<usize> = read_vec();
    for j in 0 .. tv[0] {
      sv[i].push(tv[j+1]-1);
    }
  }

  let pv: Vec<usize> = read_vec();

  let mut ans: usize = 0;
  
  for l in 0 .. (1 << n) {
    let mut f: bool = true;
    for i in 0 .. m {
      let mut p = 0;
      for j in 0 .. sv[i].len() {
        if l & (1 << sv[i][j]) != 0 { p += 1; }
      }
      if p % 2 != pv[i] { f = false; break; }
    }
    if f {ans += 1;}
  }
  
  println!("{}", ans);
}

fn read_vec<T>() -> Vec<T>
  where T: std::str::FromStr,
        T::Err: std::fmt::Debug
{
  let mut buf = String::new();
  std::io::stdin().read_line(&mut buf).expect("failed to read");
  buf.split_whitespace().map(|e| e.parse().unwrap()).collect()
}