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
n = gets.to_i
ds = gets.split.map(&:to_i)

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

h = Hash.new(0)
mx = -1
ds.each do |d|
  h[d] += 1
  mx = d if d > mx
end

ans = 0
0.upto(mx).each do |i|
  if i == 0
    if h[0] != 1
      puts 0
      exit
    end
  elsif i == 1
    ans = 1
  else
    ans *= (h[i - 1] ** h[i])
    ans %= mod
  end
end
p ans