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: Accepted)

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

Switch = Struct.new(:state) do
  def turn_on
    self.state = 'on'
  end

  def turn_off
    self.state = 'off'
  end

  def on?
    state == 'on'
  end
end

switches = n.times.each_with_object({}) do |i, hash|
  hash[i+1] = Switch.new
end

Light = Struct.new(:sw_ids, :p, :switches) do
  def on?
    sw_ids.count { |sw_id| switches[sw_id].on? } % 2 == p
  end
end

lights = []
m.times do |i|
  inputs = gets.split(' ').map(&:to_i)
  sw_ids = inputs[1..inputs[0]]
  lights << Light.new(sw_ids, nil, switches)
end

gets.split(' ').map(&:to_i).each_with_index do |p, i|
  lights[i].p = p
end

patterns = 0
b = 2**n - 1
loop do
  bits = b.to_s(2)
  order = sprintf("%0#{n}d", bits)
  order.split('').each_with_index do |o, i|
    sw = switches[i+1]
    if o == "1"
      sw.turn_on
    else
      sw.turn_off
    end
  end

  if lights.all?(&:on?)
    patterns += 1
  end

  if b == 0
    break
  end
  b -= 1
end

print patterns