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 PHP7 (7.0.15) (judge verdict: Accepted)

<?php

fscanf(STDIN, "%d %d", $n, $m);
$k_arr = [];
for ($i = 0; $i < $m; ++$i) {
    $input = explode(' ', trim(fgets(STDIN)));
    $k = $input[0];
    $k_arr[] = $k;
    for ($j = 0; $j < $k; ++$j) {
        $s_arr[$i][] = $input[$j + 1];
    }
}
$p_arr = explode(' ', trim(fgets(STDIN)));

$count = 0;
for ($switch = 0; $switch < (1 << $n); ++$switch) {

    $flg = true;
    // 電球
    for ($i = 0; $i < $m; ++$i) {

        $on_count = 0;
        // 電球にくっつくスイッチ
        foreach ($s_arr[$i] as $s) {
            if ($switch & (1 << ($s - 1))) {
                ++$on_count;
            }
        }

        if ($on_count % 2 != $p_arr[$i]) {
            $flg = false;
            break;
        }
    }

    if ($flg) {
        ++$count;
    }
}

echo $count;