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
 
//N M
list($n,$m) = explode(' ', trim(fgets(STDIN)));
 
 
//k s...s
for($i=1; $i<=$m; $i++){
  $t = explode(' ', trim(fgets(STDIN)));
  for($j=1;$j<=$t[0]; $j++){
    $l[$i][$j] = $t[$j];
  }
}
 
 
//p...p
$p = explode(' ', trim(fgets(STDIN)));
array_unshift($p,"");
 
 
//bit全検索 スイッチの状態の数だけループ
$ans = 0;
for($i=0; $i<2**$n; $i++){
  
  //スイッチの状態 (ex.00 01 10 11)
  $ls = str_pad(decbin($i), $n, 0, STR_PAD_LEFT);
  
  //それぞれのスイッチの状態で電球の数だけループ
  $c = 0;
  for($j=1; $j<=$m; $j++){
    
    //電球に繋がったスイッチの数だけループ
    $on = 0;
    foreach($l[$j] as $s){
      //繋がったスイッチがオンになっている場合にカウント
      if ($ls[$s-1] == 1) $on++;
    }
    //オンのスイッチの総数を２で割った余りが 対応するpと一致する電球の数をカウント
    if ($on%2==$p[$j]) $c++;    
  }
  //余りがpと一致した電球の数が 電球の総数と一致したらカウント
  if ($c == $m) $ans++;
  
}
 
echo $ans;
 
 
  
?>