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: Wrong Answer)

import Data.List

g :: [Int] -> Mint
g c = product $ zipWith (^) (map M c) (tail c)

f :: [Int] -> [Int]
f d = map (pred.length) $ group $ sort $ [0..maximum d] ++ d

main :: IO ()
main = getLine >> getLine >>= print.g.f.map read.words

modulus = 998244353
newtype Mint = M Int
instance Num Mint where
  (M a) + (M b)
    | a + b < modulus = M $ a + b
    | otherwise = M $ a + b - modulus
  (M a) - (M b)
    | a - b < 0 = M $ modulus + a - b
    | otherwise = M $ a - b
  (M a) * (M b) = toEnum $ a * b
  fromInteger = toEnum . fromInteger
  abs = id
  signum = const $ M 1
instance Enum Mint where
  fromEnum (M a) = a
  toEnum a = M $ mod a modulus
instance Show Mint where
  show = show . fromEnum