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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

import Control.Monad
import qualified Data.Map as Map
import Data.List

f2::Int->[Int]->Map.Map Int [Int]->Map.Map Int [Int]
f2 _ [] ms=ms
f2 n (e:es) ms=f2 n es (Map.insertWith (++) e [n] ms)

f::Int->[[Int]]->Map.Map Int [Int]->Map.Map Int [Int]
f _ [] ms=ms
f n (e:es) ms=f (n+1) es (f2 n e ms)

g2::Int->[Int]->[Int]->[Int]
g2 _ _ [] =[]
g2 no [] (x:xs)=x:g2 (no+1) [] xs 
g2 no (t:ts) (x:xs)
          |no==t=(x+1):(g2 (no+1) ts xs)
          |otherwise=x:(g2 (no+1) (t:ts) xs)

g::[[Int]]->[Int]->[Int]->Int
g [] xs ans=foldl1 (*) $ [if (mod x 2)==(mod y 2) then 1::Int else 0::Int|(x,y)<-(zip xs ans)]
g (e:es) xs ans=(g es xs ans)+(g es (g2 1 e xs) ans)


main = do
       t1<-getLine
       let n:m:[]=map read (words t1)::[Int]
       ts<-replicateM m getLine
       ks<-getLine
       let ts2=map tail $ map (map read . words) ts::[[Int]]
           ks2=map read (words ks)::[Int]
           ts3=map sort $ Map.elems $ f 1 ts2 (Map.fromList [])
       print $ (2^(n-(length ts3))) * ( g ts3 [0|e<-[1..m]] ks2)