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 PyPy3 (2.4.0) (judge verdict: Accepted)

import collections

def main():
    N=int(input())
    if N==0:
        return 0
    DL=[int(x) for x in input().split()]
    if DL[0]!=0:
        return 0
    
    
    dic=collections.Counter(DL)
    if dic[0]!=1:
        return 0
    keys=dic.keys()
    result=1
    for i in range(1,max(keys)+1):
        if dic[i]==0 and dic[i-1]==0:
            return 0
        result*=dic[i-1]**dic[i]

    return result%998244353


print(main())