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: Wrong Answer)

import sys
import collections
def solve():
    sys.setrecursionlimit(100000)
    mod = 998244353

    N = int(input())
    D = list(map(int, input().split()))
    cnter = collections.Counter(D)
    cnter = sorted(cnter.items())

    ans = 1
    if cnter[0][0] != 0 or cnter[0][1] != 1 : 
        ans = 0
    elif cnter[1][0] != 1:
        ans = 0
    else:
        for i in range(2, len(cnter)):
            if cnter[i][0] != i:
                ans = 0
                break

            else:
                ans *= cnter[i-1][1] ** cnter[i][1] % mod
    
    print(ans)


if __name__ == '__main__':
    solve()