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: Accepted)

"use strict";
function Main(input) {
  const args = input.split('\n');
  const NM = args.shift().split(' ').map(m => parseInt(m));
  const N = NM.shift();
  const M = NM.shift();
  const s = [];
  let idx = 0;
  while (args.length > 2) {
    const ks = args.shift().split(' ').map(m => parseInt(m));
    ks.shift();
    ks.map(m => m - 1).forEach(f => {
      s[f] = !!s[f] ? s[f] : [];
      s[f].push(idx);
    });
    idx++;
  }
  const p = args.shift().split(' ').map(m => parseInt(m));
  let count = calc(true, s, p, emptyArray(M), N - 1);
  count += calc(false, s, p, emptyArray(M), N - 1);
  console.log(count);
}

function calc(bool, s, p, Ms, i) {
  if (bool && !!s[i]) {
    s[i].forEach(bulb => Ms[bulb]+= 1);
  }
  if (i > 0) {
    let count = calc(true, s, p, [].concat(Ms), i - 1);
    count += calc(false, s, p, [].concat(Ms), i - 1);
    return count;
  }
  return Ms.filter((f, i) => f % 2 === p[i]).length === Ms.length ? 1 : 0;
}

function emptyArray(M) {
  const arr = [];
  for(let i = 0; i < M; i++) {
    arr[i] = 0;
  }
  return arr;
}

Main(require("fs").readFileSync("/dev/stdin", "utf8"));