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: Runtime Error)

MOD = 998244353
io = STDIN
n=io.gets.to_i
ar=io.gets.split.map(&:to_i)
hash={}
(1..ar.max).each{|i|hash[i]=0}
ar[1..n].each do |a|
  hash[a]+=1
end
def modpow(a,b)
  return 1 if b==0
  b.odd? ? (modpow(a,b-1)*a)%MOD : modpow(a**2%MOD,b/2)
end

def calc(cnts)
  rtn=1
  cnts.each_cons(2) do |a,b|
    rtn*=modpow(a,b)
  end
  rtn
end
if ar[0]!=0 || hash.values.include?(0)
  puts 0
else
  puts calc(hash.values)
end