Problem:
Score : 300 points

Problem Statement

There are N people, conveniently numbered 1 through N.
They were standing in a row yesterday, but now they are unsure of the order in which they were standing.
However, each person remembered the following fact: the absolute difference of the number of the people who were standing to the left of that person, and the number of the people who were standing to the right of that person.
According to their reports, the difference above for person i is A_i.

Based on these reports, find the number of the possible orders in which they were standing.
Since it can be extremely large, print the answer modulo 10^9+7.
Note that the reports may be incorrect and thus there may be no consistent order.
In such a case, print 0.

Constraints

1≦N≦10^5

0≦A_i≦N-1

Input

The input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the number of the possible orders in which they were standing, modulo 10^9+7.

Sample Input 1

5
2 4 4 0 2

Sample Output 1

4

There are four possible orders, as follows:

2,1,4,5,3

2,5,4,1,3

3,1,4,5,2

3,5,4,1,2

Sample Input 2

7
6 4 0 2 4 0 2

Sample Output 2

0

Any order would be inconsistent with the reports, thus the answer is 0.

Sample Input 3

8
7 5 1 1 7 3 5 3

Sample Output 3

16

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-#LANGUAGE FlexibleContexts#-}
import Control.Monad
import Control.Monad.State
import Control.Monad.Reader
import Control.Arrow
-- import Control.Comonad
import Data.Maybe
import Data.List
import Data.Bool
import Data.Fixed
import Data.Tuple
import Data.Ratio
import qualified Data.Map.Strict as M
import qualified Data.Array.IO as IA
import qualified Data.Vector as V

getI = read <$> getLine :: IO Int
getIs = map read . words <$> getLine :: IO [Int]
getD = read <$> getLine :: IO Double
getDs = map read . words <$> getLine :: IO [Double]
getInt = read <$> getLine :: IO Integer
getInts = map read . words <$> getLine :: IO [Integer]

type Row = Integer
type Col = Integer
type PointI = (Integer, Integer)
type Point = (Int, Int)

mmm = 1000000007
fact n
    | n < 1 = 1
    | True = foldr1 (*) [1..n]
ti = toInteger
fi = fromInteger
fig = fromIntegral
rf = realToFrac

(.<.) :: Integer -> Integer -> Integer
(.<.) f g -- combination
    | g < 1 || f < 1 || f < g = 1
    | True = foldr1 (*) (take (fi g) [f,f-1..]) `div` fact g
(.>.) f g --combination with repetition
    | g < 1 || f < 1 = 1
    | True = fact (f + g - 1) `div` (fact (f - 1) * fact g)
comb = (.<.)
combWp = (.>.)

unique a = unique_ (reverse . sort $ a) [] Nothing
    where
        unique_ [] arr _ = arr
        unique_ (x:xs) arr n
            | n == Just x = unique_ xs arr n
            | otherwise = unique_ xs (x:arr) $ Just x
 
zeroume keta num
    | (length . show $ num) > keta = show num
    | otherwise = do 
        let zeros = take keta $ repeat '0'
        reverse . take keta $ (reverse . show $ num) ++ zeros
 
i2b num
    | num < 2 = show num
    | otherwise = i2b_ num []
        where
            i2b_ num str
                | num == 0 = str
                | num == 1 = '1':str
                | otherwise = do
                    let next = num `div` 2
                    let this = if odd num then '1' else '0'
                    i2b_ next (this:str)

mapIncr = mapPlus 1
mapPlus n Nothing = Just n
mapPlus n a = (+) n <$> a

countIf f = length . filter f

enumDivisor :: Int -> [Int]
enumDivisor n = do
    let sqrted = floor $ sqrt (fromIntegral n)
        smaller = filter (\x -> mod n x == 0) [1..sqrted]
        bigger = reverse $ map (div n) smaller
    if head bigger == sqrted then smaller ++ (init bigger)
    else smaller ++ bigger

safeHead [] = Nothing
safeHead (x:_) = Just x

-- 
--
--

main = do
    num <- getI
    humans <- getIs
    let counted = evalState (cnt humans) M.empty
        checks = if even num then [1,3..(num - 1)] else [0,2..(num - 1)]
        able = all (valid counted) checks
    putStrLn . show $ bool 0 (foldr (\x y -> mod (x * y) mmm) 1 $ take (div num 2) [2,2..]) able
    where
        valid mp 0 = M.lookup 0 mp == Just 1
        valid mp n = M.lookup n mp == Just 2


cnt [] = get
cnt (x:xs) = do
    nowMap <- get
    put $ M.alter mapIncr x nowMap
    cnt xs