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: Wrong Answer)

var lines = [];

var readline = require('readline')
var rl = readline.createInterface({
    input: process.stdin,
    output: process.stdout,
});


rl.on('line', (x) => {
    lines.push(x)
});

rl.on('close', () => {
    var N = Number(lines[0].split(" ")[0])
    var M = Number(lines[0].split(" ")[1])
    var ans = 0;
    lines.shift()

    // 電球と繋がっているスイッチの関係性を保存
    var a = [];
    for (var i = 0; i < M; i++) {
        var t = lines[i].split(" ");

        // i番目の電球と繋がっているスイッチをa[i]に格納
        for (var j = 1; j <= Number(t[0]); j++) {
            a[t[j] - 1] |= 1 << i;
        }
    }
    
    var p = 0; var t = lines[M].split(" ");

    for (var i = 0; i < M; i++) {
        var x = Number(t[i]);
        p |= x << i;
    }

    // 電球on or 電球off全パターンを検索
    // 000,001,010,011, ... ,111
    for (var index = 0; index < (1 << N); index++) {
        var t = 0;

        // スイッチ
        for (i = 0; i < M; i++) {
            // indexのiビット目が1か判定
            if (index >> i & 1) {
                t ^= a[i]
            }
        }
        if (t === p) { ans++ }
    }
    console.log(ans)
});