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 (GNU Fortran 9.2.1) (judge verdict: Accepted)

      PROGRAM piyo
      IMPLICIT NONE
      integer :: n,m,p(10),k(10),sNum(10,10),ans=0
      logical :: s(10,10) = .false.,switch(10)
      integer :: i,j
      
      read*,n,m
      do i = 1,m
        read(fmt='(i2)',advance='no',unit=5) k(i)
        read*,sNum(i,1:k(i))
      end do
      read*,p(1:m)
      
      do i = 1,m
        do j = 1,k(i)
          s(i,sNum(i,j)) = .true.
        end do
      end do
 ! do i = 1,m
 !   print*,s(i,:)
 ! end do
      !s(n,k(i))
      
      
      outer:do i = 0,2**n-1
        switch = .false.
        do j = 1,n
          switch(j) = btest(i,j-1)
        end do
        
        do j = 1,m
          if(p(j)/= mod( count(switch(1:n).and.s(j,1:n)),2 ))then
            cycle outer
          end if
        end do
        ans = ans + 1
      end do outer
      
      print*,ans
      
 !debugg
  !    print*,n,m
  !    do i = 1,m
  !      print*,k(i),sNum(i,1:k(i))
  !    end do
  !    print*,p(1:m)
      
      END PROGRAM