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)

def AreAllTheLightsOn(SwitchPattern,M,LineVolume,Destinations,p):
    # print(SwitchPattern)
    Light=0
    for m in range(M):
        OnSwitch=0
        for l in range(LineVolume[m]):
            a=Destinations[m][l]-1
            if SwitchPattern[a]==1:
                OnSwitch += 1
            else:
                pass
        if OnSwitch%2==p[m]:
            Light+=1
    # print (Light)
    if Light==M:
        return True


N,M=map(int,input().split())
K=[]
for _a in range (M):
    arr=list(map(int,input().split()))
    K.append(arr)
LineVolume=[] #m番目の電球がスイッチとつながっている線の数
Destinations=[] #m番目の電球がつながっているスイッチの番号(1スタート)
for _b in range (M):
    LineVolume.append(K[_b][0])
    Destinations.append(K[_b][1:])
p=list(map(int,input().split()))
# print (LineVolume)
# print (Destinations)
# print(p)

LightOnPattern=0

for Pattern in range (2**N):
    SwitchPattern=[]
    for ONOFF in range(N):
        if(Pattern>>ONOFF)&1==1:
            SwitchPattern.append(1)
        else:
            SwitchPattern.append(0)
    # print(SwitchPattern)
    if AreAllTheLightsOn(SwitchPattern,M,LineVolume,Destinations,p)==True:
        LightOnPattern += 1

print(LightOnPattern)