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: Runtime Error)

N = gets.chomp.to_i
D = gets.chomp.split(" ").map(&:to_i)

if D[0] != 0 || D[1..-1].include?(0) then
  puts "0"
  return
end

ranks = []
(1..Float::INFINITY).lazy.each do |i|
  r = D.count {|n| n == i}
  break if r == 0
  ranks << r
end

$memo = []
def factorial(n)
  $memo[n] if $memo[n] != nil

  $memo[n] = (1..n).inject(1,:*)
  return $memo[n]
end

def C(n, r)
  return factorial(n) / (factorial(n) / factorial(n - r))
end

s = 1
ranks.each do |r|
  (0..r).each do |n|
    s *= C(r, n)
  end
end

puts s % 998244353