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'

class Main {
  solve() {
    const nm = input.nextIntArr();
    const n = nm[0], m = nm[1];
    const ksList = input.nextIntRange(m);
    const p = input.nextIntArr();

    const bitCount = (n) => {
      n = n - ((n >> 1) & 0x55555555)
      n = (n & 0x33333333) + ((n >> 2) & 0x33333333)
      return ((n + (n >> 4) & 0xF0F0F0F) * 0x1010101) >> 24
    }

    const pattern = [];
    for (const ks of ksList) {
      ks.shift();
      pattern.push(_.sum(ks.map(e=>1<<(e-1))));
    }
    return _.range(1<<n).map((i)=>pattern.every((e,j)=> (bitCount(e&i)%2)===p[j])).filter(e=>e).length;
  }
}

class Mod {
  constructor(mod) {
    if (mod == null) { mod = Combinatorics.DefaultMod; }
    this.mod = mod;
  }
  mult(a,b) {
    let ret = 0;
    a %= this.mod;
    b %= this.mod;
    while (b) {
      if (b&1) {
        ret += a;
        if (ret > this.mod) { ret -= this.mod; }
      }
      a <<= 1;
      if (a > this.mod) { a -= this.mod; }
      b >>= 1;
    }
    return ret;
  }
  pow(a,b) {
    let ret = 1;
    while(b) {
      if (b&1) { ret = this.mult(ret,a); }
      a = this.mult(a,a);
      b >>= 1;
    }
    return ret;
  }
  div(a,b) { return this.mult(a, this.pow(b, this.mod-2)) }
  comb(n,k) { return MOD.mult(factTable[n],MOD.mult(invFactTable[k],invFactTable[n-k])); }
}
const MOD = new Mod(1e9+7);

class Underscore {
  sum(arr) { return arr.reduce((a,b)=>a+b); }
  modSum(arr,mod) { if (mod == null) { mod = MOD.mod; } return arr.reduce((a,b)=>(a+b)%mod); }
  min(arr, func) { if (func == null) {func = (a,b) => a - b} return arr.reduce((a,b) => func(a,b) < 0 ? a : b); }
  max(arr, func) { if (func == null) {func = (a,b) => a - b} return arr.reduce((a,b) => func(a,b) > 0 ? a : b); }
  sort(arr, func) { if (func == null) {func = (a,b) => a - b} return arr.sort(func) }
  uniq(arr) { 
    arr.sort();
    for(let i = 0; i < arr.length - 1; ++i) { 
      arr[i] = arr[i+1] == arr[i] ? null : arr[i]; 
    }
    return arr.filter(e => e != null);
  }
  range(start, stop, step) {
    if (stop == null) { stop = start || 0; start = 0; }
    if (!step) { step = stop < start ? -1 : 1; }
    const length = Math.max(Math.ceil((stop - start) / step), 0);
    const range = Array(length);
    for (var idx = 0; idx < length; idx++, start += step) { range[idx] = start; }
    return range;
  }
  first (array, n) {
    if (array == null || array.length < 1) return n == null ? void 0 : [];
    if (n == null) return array[0];
    return this.initial(array, array.length - n);
  };
  initial (array, n) { return array.slice.call(array, 0, Math.max(0, array.length - (n == null ? 1 : n))); }
  tail(array, n) { return array.slice.call(array, n == null ? 1 : n); }
  last (array, n) { if (n == null) return array[array.length - 1]; return this.tail(array, Math.max(0, array.length - n)); }
  sumArray (arr) { return this.tail(arr.reduce(((a,b,i) => {a.push(b + a[i]); return a}), [0])); }
  diffArray (arr) { const s = []; for (let i = 1; i < arr.length; ++i) { s.push(arr[i] - arr[i-1]); } return s; }
  // 配列の値以下の最大のindex
  binarySearch (array, key, iteratee) {
    const itr = (obj) => (typeof obj === 'object') ? obj[iteratee] : obj;
    const value = itr(key);
    let low = 0, high = array.length;
    while (low < high) {
      let mid = Math.floor((low + high) / 2);
      if (itr(array[mid]) < value) low = mid + 1; else high = mid;
    }
    return low;
  };
}
const _ = new Underscore();

class Input {
  constructor() {
    this.cursor = 0;
    this.inputAll();
  }

  inputAll() {
    this.data = (require("fs").readFileSync("/dev/stdin", "utf8")).split('\n');
    return this.data;
  }

  /* 1行 or 複数行文字列 */
  nextLine(n) {
    if (n) {
      const ret = this.data.slice(this.cursor, this.cursor + n);
      this.cursor += n;
      return ret;
    }
    return this.data[this.cursor++];
  }
  /* 1整数 */
  nextInt() { return parseInt(this.nextLine()); } 
  /* 一行文字配列 */
  nextStrArr() { return this.nextLine().split(' '); } 
  /* 一行整数配列 */
  nextIntArr() { return this.nextStrArr().map((e) => parseInt(e)); }
  /* 一行浮動小数点配列 */
  nextFloatArr() { return this.nextStrArr().map((e) => parseFloat(e)); }
  /* 複数行整数配列 */
  nextIntLine(n) { return this.nextLine(n).map((e) => parseInt(e)) }
  /* 複数行浮動小数点配列 */
  nextFloatLine(n) { return this.nextLine(n).map((e) => parseFloat(e)) }
  /* 複数行文字列二次元配列 */
  nextStrRange(n) { return this.nextLine(n).map((line) => line.split(' ')); }
  /* 複数行整数二次元配列 */
  nextIntRange(n) {
    return this.nextLine(n).map((line) => line.split(' ').map((e) => parseInt(e)));
  }
  /* 複数行浮動小数点数二次元配列 */
  nextFloatRange(n) {
    return this.nextLine(n).map((line) => line.split(' ').map((e) => parseFloat(e)));
  }
} 
const input = new Input();

class Out {
  constructor() {
    this.option = {
      'true': 'Yes',
      'false': 'No',
    };
  }
  write(output){
    let ret;
    if (output == null) return;
    if (output === true) { ret = this.option['true']; }
    else if (output === false) { ret = this.option['false']; }
    else { ret = output; }
    console.log(ret);
  }
  bool(option) {
    this.option['false'] = option[0];
    this.option['true'] = option[1];
    return this;
  }
}
const out = new Out();

out.write(new Main().solve());