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 Fortran (gfortran v4.8.4) (judge verdict: Accepted)

PROGRAM ATCODER

implicit none
integer(16) :: n,m,i,max=0,l,count=0,h,j,on,check=0
integer(16),allocatable :: k(:),s(:,:),p(:),sw(:)

read*, n,m
allocate(k(m),s(m,10),p(m),sw(n))
s = 0
do i = 1, m, 1
  read*, k(i),s(i,1:k(i))
end do
read*, p(1:m)

do i = 1, n
  max = max + 2**(i-1)
end do

do i = 0, max, 1
  check = 0
  l = i
  sw = 0
  do j = 1, n, 1
    sw(j) = mod(l,2)
    l = l/2
  end do
  do j = 1, m, 1
    on = 0
    do h = 1, k(j), 1
      if ( sw(s(j,h)) == 1 ) then
        on = on + 1
      end if
    end do
    if ( mod(on,2) /= p(j) ) then
      check = 1
      exit
    end if
  end do
  if ( check == 0 ) then
    count = count + 1
  end if
end do

print'(i0)', count

END PROGRAM ATCODER