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 Ruby (2.3.3) (judge verdict: Wrong Answer)

input = gets.chomp.split
n = input[0].to_i
m = input[1].to_i

sw = []
for i in 1..m do
    sw.push(gets.chomp.split.map(&:to_i))
end

ps = gets.chomp.split(" ").map(&:to_i)

switch = Array.new(n,0)

light = true

for i in 1..m do
    if sw[i-1][0] == 1
        if ps[i-1] == 0
            if switch[i-1] == 1
                light = false
            else
                switch[i-1] = -1
            end
        else
            if switch[i-1] == -1
                light = false
            else
                switch[i-1] = 1
            end
        end
    end
end

def tansaku(i,n,m,sw,ps,switch)
    count = 0
    if i > n
        lt = true
        for j in 1..m do
            cnt = 0
            k = sw[j-1][0]
            for j2 in 1..k do
                if switch[j2-1] == 1
                    cnt += 1
                end
            end
            if cnt % 2 != ps[j-1]
                lt = false
            end
        end
        if lt 
            return 1
        else
            return 0
        end

    else
        if switch[i-1] == 0
            # print i, " = " "on\n" 
            switch[i-1] = 1
            count += tansaku(i+1,n,m,sw,ps,switch)
            # print i, " = " "off\n"
            switch[i-1] = -1
            count += tansaku(i+1,n,m,sw,ps,switch)
            switch[i-1] = 0
        else
            # print i," = ", switch[i-1],"\n"
            count += tansaku(i+1,n,m,sw,ps,switch)
        end
    end
    return count
end 

if light
    puts tansaku(1,n,m,sw,ps,switch)
else
    puts "0"
end