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)

# util functions

# debug
$debugFlag = ( ARGV[0] != nil )
p "==== DEBUG MODE ====" if $debugFlag

def pd(str)
  p str if $debugFlag
end

# input
inputs = []
lc = 0
while l = $stdin.gets do
  if false
    inputs << l.chomp.split(" ").map(&:to_s)
  else
    inputs << l.chomp.split(" ").map(&:to_i)
  end
  lc += 1
end

# input check
if $debugFlag
  p "--- INPUT (begin) ---"
  inputs.each do |l_i|
    p l_i
  end
  p "--- INPUT (end) ---"
  print "\n"
end

# body
if $debugFlag
end

nn = inputs[0][0]
print "nn:" if $debugFlag
pd nn
consts = []
for i in 1..inputs.size-2 do
  consts << [inputs[i][1..-1], inputs[-1][i-1]]
end

pd consts

cnt = 0
for i in 0..2**nn-1 do
  pd "i=#{i}"
  flag = true
  for j in 0..consts.size-1 do
    pd "j=#{j}"
    tmp = 0
    tmpstr = ""
    consts[j][0].size.times do |k|
      tmp += i[consts[j][0][k]-1]
      tmpstr += i[consts[j][0][k]].to_s + "+"
    end
    pd tmpstr
    if (tmp % 2) != consts[j][1]
      flag = false
      break
    end
  end
  if flag == true
    pd "i=#{i}: true"
    cnt += 1
  else
    pd "i=#{i}: false"    
  end
end

puts cnt