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.chomp.to_i
d = Array.new(N)
e = Array.new(N,0)
max = 0
str = gets.split
for i in 0...N do
    d[i] = str[i].to_i
    x = d[i]
    max = max < x ? x : max
    e[x] += 1
end
if d[0] != 0 || e[0] != 1
    puts 0
    exit
end
answer = 1
for i in 2..max do
    x = e[i]
    y = e[i-1]
    while x > 0
        answer *= y
        x -= 1
    end
    answer %= 998244353
end
puts answer