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)

MOD = 998244353

def main(argv)
  n = gets.chomp.to_i
  d = gets.chomp.split(' ').map(&:to_i)

  unless d[0] == 0 then
    puts 0.to_s
    return
  end

  c = Hash.new
  d.each do |x|
    if c.has_key?(x) then
      c[x] += 1
    else
      c[x] = 1
    end
  end

  unless (c.has_key?(0) and (c[0] == 1)) then
    puts 0.to_s
    return
  end

  m = c.keys.max

  s = 1
  h = 0
  while h < m do
    unless (c.has_key?(h) and c.has_key?(h + 1)) then
      puts 0.to_s
      return
    end

    t = (c[h] ** c[h + 1]) % MOD
    s = (s * t) % MOD

    h += 1
  end

  puts s.to_s
end

if self.to_s == 'main' then
  main(ARGV)
end