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)

n, m = gets.split.map(&:to_i)

$bulbs = []
cnt = 0
$res = 0
$set = []

def bulb_on?(bulb, condition, switches)
    n = 0
    bulb.each { |i| n += switches[i] }
    return (n % 2) == condition
end

def solve(switches, idx, conditions)
    [0, 1].each do |i|
        switches[idx] = i unless switches[idx].nil?
        unless $set.include?(switches)
            all_on = true
            $bulbs.each_with_index do |b, bi|
                unless bulb_on?(b, conditions[bi], switches)
                    all_on = false
                    break
                end
            end
            $res += 1 if all_on
            $set << switches.dup
        end
        solve(switches, idx + 1, conditions) unless idx >= switches.size
    end
end

candidates = {}

m.times do |i|
    inp = gets.split.map(&:to_i)
    inp.shift
    $bulbs.push(inp.map { |i| i-1 })
end

$bulbs.each { |i| cnt = [cnt, i.max].max }
switches = Array.new(cnt + 1, 0)
switches = [] if cnt == 0
conditions = gets.split.map(&:to_i)
solve(switches, 0, conditions)

puts $res