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.split.map(&:to_i)
bulbs = {}
m.times do |i|
  tmp = gets.split.map(&:to_i)
  tmp.shift
  bulbs[i+1] = { switches: tmp }
end
ps = gets.split.map(&:to_i)

ps.each.with_index(1) do |p, i|
  bulbs[i][:p] = p
end

sw = {}
n.times do |i|
  sw[i+1] = false
end

ans = 0
(0..("1" * n).to_i(2)).each do |i|
  bit = i.to_s(2).rjust(n, '0')
  on_off = bit.chars.map{|j| j == '1' }

  on_off.each.with_index(1) do |bool, i|
    sw[i] = bool
  end

  flg = true
  m.times do |j|
    on_count = bulbs[j+1][:switches].count{|s| sw[s]}
    if on_count % 2 != bulbs[j+1][:p]
      flg = false
    end
  end

  ans += 1 if flg
end

puts ans