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)

def ppow(x, n)
    return 1 if n == 0
    a = 1
    n.times do
        a = (a * x) % 998244353
    end
    return a
end

n = gets.to_i
d = gets.split.map(&:to_i)

if d[0] != 0 || d[1..-1].include?(0)
    puts 0
    exit
end

c = []
max = 0
d.each do |x|
    a = c[x] || 0
    c[x] = a + 1
end

if c.include?(nil)
    puts 0
    exit
end

ans = 1
prev = 1
c.each do |n|
    ans *= ppow(prev, n)
    ans %= 998244353
    prev = n
end

puts ans