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)

input = require('fs').readFileSync('/dev/stdin', 'utf8')
lines = input.split('\n')
line = lines.shift().split(' ')
N = line[0] * 1
M = line[1] * 1
k = []
for(i=0; i<M ; i++) {
    k[i] = lines[i].split(' ').slice(1).map((x) => x * 1)
    arr = new Array(N).fill(0)
    for (j=0;j<k[i].length;j++) {
        arr[N - k[i][j]] = 1
    }
    k[i] = parseInt(arr.join(''),2)
}
p = lines[i].split(' ').map((x) => x * 1)
max = Math.pow(2, N)
cnt = 0
for (i=0; i<max; i++) {
    bool = true
    for(j=0; j<M ; j++) {
        t = (i & k[j]).toString(2)
        c = 0
        for (s = 0; s < t.length; s++) {
            if (t[s] == 1) {
                c++
            }
        }
        bool = bool && (c % 2 == p[j])
    }
    if (bool) cnt++
}

console.log(cnt)