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)

# スイッチ数, 電球数
n, m = gets.chomp.split(' ').map(&:to_i)

def ep(o)
  STDERR.puts o.inspect
end

def to_a(array)
  result = 0
  for i in 0..array.reverse.length-1
    result = result << 1
    result += array[i]
  end
  result
end

sls = Array.new(n){|i| Array.new(m, 0)}
for i in 0..m-1 do
  _, *ss = gets.chomp.split(' ').map(&:to_i)
  for s in ss do
    sls[s-1][i] = 1
  end
end

ep sls.inspect

dp = Array.new(2**m, 0)
dp[0] = 1
ep dp

for sl in sls do
  dp2 = dp.clone
  sum = 0
  ep sl
  sum = to_a(sl)
  ep sum
  # OFFの場合
  #   -> なにもしない
  # ONの場合
  for x in 0..dp.length-1 do
	dp2[sum^x] += dp[x]
  end
  dp = dp2
  ep dp
end
ps = gets.chomp.split(' ').map(&:to_i)

ep to_a(ps)
print dp[to_a(ps)]
print "\n"