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 JavaScript (node.js v5.12) (judge verdict: Runtime Error)

"use strict";
    
function permutateWithRepetitions(
  permutationOptions,
  permutationLength = permutationOptions.length
) {
  if (permutationLength === 1) {
    return permutationOptions.map(permutationOption => [permutationOption]);
  }

  // Init permutations array.
  const permutations = [];

  // Get smaller permutations.
  const smallerPermutations = permutateWithRepetitions(
    permutationOptions,
    permutationLength - 1
  );

  // Go through all options and join it to the smaller permutations.
  permutationOptions.forEach((currentOption) => {
    smallerPermutations.forEach((smallerPermutation) => {
      permutations.push([currentOption].concat(smallerPermutation));
    });
  });

  return permutations;
}

const main = arg => {
    arg = arg.trim().split("\n");
    const N = parseInt(arg[0].split(" ")[0]);
    const M = parseInt(arg[0].split(" ")[1]);
    
    const info = arg.slice(1, M + 1);
    const p = arg.slice(-1)[0].split(" ").map(n=>parseInt(n));
    
    const combos = permutateWithRepetitions([0, 1], N);
    let answer = 0;

    for(let i in combos) {
        let lightOn = 0;

        for(let j=0; j<M; j++) {
            let temp = info[j].split(" ").map(n=>parseInt(n));
            let k = temp[0];
            let s = temp.slice(1, k + 1);
            
            let cnt = 0;
            
            for(let k=0; k<s.length; k++) {
                if(combos[i][s[k] - 1] === 1) cnt++;
            }

            if(cnt % 2 === p[j]) {
                lightOn++;
            }
        }
        
        if(lightOn === M) {
            answer++;
        }
    }
    
    console.log(answer);
}
main(require('fs').readFileSync('/dev/stdin', 'utf8'));