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 Python (3.4.3) (judge verdict: Wrong Answer)

import sys
N = int(input())
D = list(map(int, input().split()))

Ans = 0
number = 1
n_count = 0
b_count = 1

newD = sorted(D)
if N == 1:
  Ans = 1
  print(Ans)
  sys.exit()

if newD[0] != 0 or newD[1] != 1:
  print(Ans)
  sys.exit()
else:
  
  for i in range(1,N):
    if newD[i] == number:
      n_count = n_count + 1
    elif newD[i] == number + 1:

      if number == 1:
        Ans = 1
        #print(Ans)
      else:
        Ans = Ans * (b_count ** n_count)
        if Ans > 998244353:
          Ans = Ans % 998244353
        #print(Ans)
        
      number = number + 1
      b_count = n_count
      n_count = 1
    else:
      Ans = 0
      print(Ans)
      sys.exit()

  if number == 1:
    Ans = 1
  else:
    Ans = Ans * (b_count ** n_count)
    Ans = Ans % 998244353
  print(Ans)