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: Wrong Answer)

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

ans = 1
if D[0] != 0 then
  ans = 0
elsif (y = D.sort.bsearch_index{|x| x >= N - 1}) && y < N - 1 then
  ans = 0
else
  j = 1
  for i in 1..(N-1) do
    if D[i] == 0 then
      ans = 0
      break
    elsif i == N - 1 then
      ans = ans * D[j..i].min
      break
    elsif D[i] >= D[i+1] then
      p D[j..i].min
      ans = ans * D[j..i].min
      j = i + 1
    end
  end
end

p ans.modulo(998244353)