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 Lua (5.3.2) (judge verdict: Runtime Error)

local n=io.read("n")
local verte={}
local flag=true
for i=1,n do
    local x=io.read("n")
    if (i==1 and x>0) or (i>1 and x==0) then
        flag=false
    end
    verte[x]=(verte[x] or 0)+1
end

if not flag then
    print(0)
else
    local tree=1
    local mod=998244353
    for x,_ in pairs(verte) do
        if x+1<=#verte then
            if not verte[x] or not verte[x+1] then
                print(0)
                return
            end
            tree=tree*(verte[x]^verte[x+1])%mod
        end
    end
    print(string.format("%d",tree))
end