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.Exception (assert)
import           Control.Monad     (replicateM)

main :: IO ()
main = do
  [n, m] <- map read . words <$> getLine
  sss <- replicateM m $ do
    k : ss <- map read . words <$> getLine
    assert (length ss == k) $
      pure ss
  ps <- map read . words <$> getLine
  assert (length ps == m) $
    print $ solve n $ zip ps sss

type Switch = Int
type Parity = Int
type BulbSetting = (Parity, [Switch])
type Bulb = [Bool] -> Bool

solve :: Int -> [BulbSetting] -> Int
solve n bulbSettings =
  length $
  filter id $
  map (\state -> all ($ state) bulbs) allStates
  where
    allStates = replicateM n [False, True]
    bulbs = map makeBulb bulbSettings

makeBulb :: BulbSetting -> Bulb
makeBulb (parity, switches) state =
  length (filter id (takes (map pred switches) state)) `mod` 2 == parity

takes :: [Int] -> [a] -> [a]
takes []     _      = []
takes _      []     = []
takes (0:is) (x:xs) = x : takes (map pred is) xs
takes is     (_:xs) = takes (map pred is) xs