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 (2.7.1) (judge verdict: Accepted)

n, m = gets.chomp.split.map(&:to_i)
k_list_list = (1..m).map do
  gets.chomp.split.map(&:to_i)
end

p_list = gets.chomp.split.map(&:to_i)


def main(n, m, k_list_list, p_list)

  ans = 0;
  (0..2**n - 1).each do |n_ten|
    n_two = n_ten.to_s(2).rjust(n, '0')
    all_switch_statuses = n_two.chars.map(&:to_i)

    light_statuses = (0..m-1).map do |light_index|
      switch_indexes = k_list_list[light_index][1..]
      is_on = switch_indexes.map{ |index| 
        all_switch_statuses[index - 1] 
      }.sum % 2 === p_list[light_index]
    end

    unless light_statuses.include?(false)
      ans += 1
    end
  end
  ans
end

puts main(n, m, k_list_list, p_list)


# require 'minitest/autorun'

# class FooTest < Minitest::Test
#   def test_case1
#     assert_equal(1, main(2, 2, [[2, 1, 2], [1, 2]], [0, 1]))
#   end
# end