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 Python (3.4.3) (judge verdict: Accepted)

#!/usr/bin/env python
# -*- coding: utf-8 -*-

from sys import stdin


def bit_count(x):
    return bin(x).count("1")

def bit_exchange(x):
    return 2 ** (x-1)

all_switch_list=[]
clear_list=[]


N, M = [int(x) for x in stdin.readline().rstrip().split()]

for i in range(M):
    switch_list = [int(x) for x in stdin.readline().rstrip().split()]
    bin_s_list = 0
    for j in range(switch_list[0]):
        # j+1 
        bin_s_list = bin_s_list | bit_exchange(switch_list[j+1])

    
    all_switch_list.append(bin_s_list)
remainder = [int(x) for x in stdin.readline().rstrip().split()]

#for i in range(bit_exchange(N+1)):

clear_list = list(range(bit_exchange(N+1)))
for i in range(M):
    for j in range(bit_exchange(N+1)):
        count = j & all_switch_list[i]
        if bit_count(count) % 2 != remainder[i]:
            try:
                clear_list.remove(j)
            except:
                pass

print(len(clear_list))