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 = スイッチ数, m = 電球数
n, m = gets.chomp.split(' ').map {|n| n.to_i}

k_list = [] #i番目の電球に繋がるスイッチ数
s_lists = [] #i番目の電球に繋がるスイッチのリスト
p_list = [] #i番目の電球の繋がる点灯しているスイッチの個数を2で割った余り

m.times do |i|
  line = gets.split(' ').map {|n| n.to_i}
  k = line[0]
  k_list << k
  s_lists << line.drop(1)
end
p_list = gets.split(' ').map {|n| n.to_i}

# pp k_list
# pp s_lists
# pp p_list

def check_ok?(s_case, s_lists, p_list)
  m_i = 0
  s_lists.each do |s_list|
    on_total = 0
    s_list.each do |s_id|
      on_total += s_case[s_id - 1]
    end
    return false unless on_total % 2 == p_list[m_i]
    m_i += 1
  end
  true
end

ans = 0
all_list = [0, 1].repeated_permutation(n).to_a
all_list.each do |s_case|
  ans += 1 if check_ok?(s_case, s_lists, p_list)
end

puts ans