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 PyPy3 (2.4.0) (judge verdict: Runtime Error)

N, M = map(int, input().split())

klist = []
sdict = {}
for m in range(M):
    tmp = list(map(int, input().split()))
    klist.append(tmp[0])
    sdict[m] = tmp[1:]
plist = list(map(int, input().split()))

def solve(N, plist, switch_num):
    global M
    num = 0

    if N == 1:
        idx = []
        idx2 = []
        for m in range(M):
            if switch_num in sdict[m]:
                idx.append(plist[m])
            else:
                idx2.append(plist[m])
        if sum(idx) == len(idx):
            if sum(idx2) == 0:
                return 1
            else:
                return 0
        elif sum(idx) == 0:
            if sum(idx2) == 0:
                return 2
        else:
            return 0

    # switch1 == OFF
    new_plist = plist
    num += solve(N-1, new_plist, switch_num+1)
    
    # switch1 == ON
    new_plist = []
    for i in range(M):
        if switch_num in sdict[i]:
            new_plist.append(1 - plist[i])
        else:
            new_plist.append(plist[i])
    num += solve(N-1, new_plist, switch_num+1)

    return num

print(solve(N, plist, 1))