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)

n = gets.chop.to_i
a = gets.chop.split(' ').map { |e| e.to_i }

if (a[0] != 0)
    puts 0
    exit
end    
 
max = 0
m = {}
 
a.each do |num|
    max = [max, num].max
    if m[num]
        m[num] += 1
    else
        m[num] = 1
    end
end
 
if (m[0] != 1)
    puts 0
    exit
end    
 
prevCount = 1
ans = 1
 
(1..max).each do |i|
    unless m[i]
        puts 0
        exit
    end
 
    m[i].times do
        ans = (ans * prevCount) % 998244353
    end
 
    prevCount = m[i]
end
 
puts ans