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)

#!/usr/bin/env ruby

nn, mm = gets.chomp.split.map(&:to_i)
ss = []
1.step(mm, 1) do |_i|
  k = gets.chomp.split.map(&:to_i)
  ss[_i - 1] = k[1..-1]
end
ps = gets.chomp.split.map(&:to_i)

ans = 0
0.step((2**nn) - 1, 1) do |bit2s|
  flag = true
  0.step(mm - 1, 1) do |ite_m|
    sum = 0
    ss[ite_m].each do |swi|
      next if bit2s[swi-1].to_i != 1
      sum += 1
    end
    if sum%2 != ps[ite_m]%2
      flag = false
      break
    end
  end
  ans += 1 if flag
end

p ans

=begin

# 一行に整数ひとつだけ
n = gets.chomp.to_i
p n

# 半角区切り整数を配列で受ける
# k1, k2 , ...
k = gets.chomp.split.map(&:to_i)

p k

# n
# a1, b1
# ...
# an, bn
# の形式

nn = gets.chomp.to_i
p nn
a = []
b = []
0.step(nn-1) do |i|
  k = gets.chomp.split(' ').map(&:to_i)
  a << k[0]
  b << k[1]
end

0.step(nn-1) { |i|
  puts "i: #{i} a:#{a[i]} b:#{b[i]}"
}

=end