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.to_i
d = gets.split.map(&:to_i)
if d[0] != 0
    ans = 0
else
    if n == 1
        ans = 1
    elsif n == 2
        ans = 1 if d[1] == 1
        ans = 0 if d[1] != 1
    else
        d.sort!
        c = 1
        now = 1
        prev = 1
        ans = 1
        1.upto(n-2) do |i|
            if d[i] == now
                if d[i] == d[i+1]
                    c += 1
                    ans = ans * (prev**c) if i == n-2
                else
                    ans = ans * (prev**c)
                    ans = ans * c if i == n-2 
                    prev = c
                    c = 1
                    now += 1
                end
            else
                ans = 0
                break
            end
        end
    end    
end
puts ans%998244353