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)

((stdin) => {
  const lines = stdin.trim().split('\n');
  const [n, m] = lines.shift().split(' ').map(Number);
  const lights = [];
  for (let i = 0; i < m; i++) {
    const light = lines.shift().split(' ').map(Number);
    lights.push(light.slice(1));
  }
  const mods = lines.shift().split(' ').map(Number);
  let count = 0;
  let base = '';
  for (let i = 0; i < n; i++) {
    base += '0';
  }
  for (let bit = 0; bit < Math.pow(2, n); bit++) {
    const b = (base + bit.toString(2)).slice(-n);
    let isAll = true;
    for (let i = 0; i < m; i++) {
      let onCount = 0;
      for (let j = 0; j < lights[i].length; j++) {
        onCount += Number(b[lights[i][j] - 1]);
      }
      if (onCount % 2 !== mods[i]) {
        isAll = false;
        break;
      }
    }
    if (isAll) {
      count += 1;
    }
  }
  console.log(count);
})(require('fs').readFileSync('/dev/stdin', 'utf8'));