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: Wrong Answer)

import Control.Monad

import Data.Ord
import Data.List
import Data.Function
import Data.Maybe

import qualified Data.ByteString.Char8 as BS

import Debug.Trace

tuplify2 :: [a] -> (a, a)
tuplify2 (x:y:_) = (x, y)
tuplify2 _ = undefined

--Input functions with ByteString
readInt = fst . fromJust . BS.readInt
readIntTuple = tuplify2 . map readInt . BS.words
readIntList = map readInt . BS.words

getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine
getIntNList n = map readIntList <$> replicateM (fromIntegral n) BS.getLine
getIntMatrix = map readIntList . BS.lines <$> BS.getContents
getIntTuple = readIntTuple <$> BS.getLine
getIntNTuples n = map readIntTuple <$> replicateM (fromIntegral n) BS.getLine
getIntTuples = map readIntTuple . BS.lines <$> BS.getContents

main = do
  [nSw,nLight] <- getIntList
  ss <- getIntNList nLight
  swAllOn <- getIntList
  let swMatrix = map (indexToVec nSw 1 . drop 1) ss
  print . sum . map (f swAllOn swMatrix) $ allSw nSw

indexToVec :: Int -> Int -> [Int] -> [Int]
indexToVec n i [] = take (n-i+1) [0,0..]
indexToVec n i indices@(x:xs)
  | x == i    = (1 : indexToVec n (i+1) xs)
  | otherwise = (0 : indexToVec n (i+1) indices)

f :: [Int] -> [[Int]] -> [Int] -> Int
f ps swMatrix sw
  | ps' == ps = 1
  | otherwise  = 0
  where
    ps' = map (flip mod 2 . sum . zipWith (*) sw) swMatrix

allSw :: Int -> [[Int]]
allSw 0 = [[]]
allSw n = [x:y | x <- [0,1], y <- allSw (n-1)]