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)
@lights = []
m.times do
  @lights.push(gets.split(" ").map(&:to_i))
end

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

@lights.each do |li|
  li.shift(1)
end

@switches = Array.new(@n, false)

@sum = 0

def recursive_dfs(switch_num, bool)
  if switch_num >= 0
    @switches[switch_num] = bool
  end
  if switch_num >= @n - 1
    judge = true
    switch_sum = 0
    @lights.each_with_index do |light, index1|
      light.each_with_index do |switch, index2|
        if @switches[switch - 1] == true
          switch_sum += 1
        end
      end
      if @ps[index1].even? != switch_sum.even?
        judge = false
      end
    end
    if judge == true
      @sum += 1
    end
    return
  end
  switch_num += 1
  2.times do |index|
    if index == 0
      recursive_dfs(switch_num, true)
    elsif index == 1
      recursive_dfs(switch_num, false)
    end  
  end
end

recursive_dfs(-1, false)
puts @sum