Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

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

import qualified Data.ByteString.Char8 as BS
import Data.Maybe (fromJust)
import Data.List

check1 :: [Integer] -> Bool
check1 (x:xs) = x == 0 && (length . filter (==0)) xs == 0

check2 :: [Integer] -> Bool
check2 [] = True
check2 [_] = True
check2 (x1:x2:xs) = succ x1 == x2 && check2 (x2:xs)

solve :: [Integer] -> Integer
solve [_] = 1
solve (x1:x2:xs) = (x1^x2) * solve (x2:xs)

main :: IO ()
main = do
    _ <- getLine
    d <- map (fst . fromJust . BS.readInteger)
        . BS.words <$> BS.getLine :: IO [Integer]
    let d' = group $ sort d
    print $
        if check1 d && check2 (map head d')
            then solve (map (fromIntegral . length) d') `rem` 998244353
            else 0