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)

mod = 998244353

def rec(n, p, m)
  if p == 0
    return 1
  elsif p % 2 == 0
    tmp = rec(n, p / 2, m)
    return tmp * tmp % m
  end
  n * rec(n, p - 1, m) % m
end

n = gets.chomp.to_i
arr = gets.chomp.split(' ').map(&:to_i)
counts = Array.new(n, 0)
(0...n).each do |i|
  counts[arr[i]] += 1
end
while counts.last == 0
  counts.pop
end
prev = counts.shift
if prev != 1
  puts 0
  exit(0)
end
result = 1
counts.each do |i|
  result = (result * rec(prev, i, mod)) % mod
  prev = i
end
puts result