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 Ruby (2.3.3) (judge verdict: Accepted)

require 'openssl'
require 'set'; require 'prime'
INF=Float::INFINITY; MOD=998244353
n = gets.chomp.to_i
d = gets.chomp.split.map(&:to_i)

if d[0] != 0
  puts 0
  exit
end

h = Hash.new(0)
d.each do |num|
  h[num] += 1
end

if h[0] != 1
  puts 0
  exit
end

ans = 1

ha = h.sort_by{|k,v| k}
(ha.size-1).times do |i|
  if ha[i][0] + 1 != ha[i+1][0]
    puts 0
    exit
  end
  # p [ha[i][1], ha[i+1][1], ha[i][1].to_bn.mod_exp(ha[i+1][1], MOD)]
  a = ha[i][1].to_bn.mod_exp(ha[i+1][1], MOD)
  ans = (ans * a) % MOD
end

puts ans