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: Runtime Error)

N, M = gets.chomp.split.map(&:to_i) # N個のスイッチ, Mの電球
a = []
c = []
M.times do |m|
  g = gets.split.map(&:to_i)
  a << [*g[1..-1]]
  c << g[0]
end
b = gets.split.map(&:to_i)

p N, M
pp a
pp b
pp c
puts
count = 0

comb_flag = true
[true, false].repeated_permutation(M) do |arr| # arrには, M 個の {T, F}の組み合わせ
  comb_flag = true
  puts
  p arr
  a.each_with_index do |aa, ii|

    print'aa->'; p aa
    p arr.values_at(*aa.map{|aaa|aaa-1})
    p arr.values_at(*aa.map{|aaa|aaa-1}).select{|aa|aa}
    p b[ii]
    p 'last!' if (ii == a.size - 1)
    p (arr.values_at(*aa.map{|aaa|aaa-1}).select{|aa|aa}.size % 2 == b[ii])

    if (arr.values_at(*aa.map{|aaa|aaa-1}).select{|aa|aa}.size % 2 == b[ii]) && comb_flag
      count += 1 if (ii == a.size - 1)

    else
      comb_flag = false
    end
  end
end

puts "こたえ"
p count