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 Python (3.8.2) (judge verdict: Wrong Answer)

def main():
    import sys
    input = sys.stdin.readline
    n = int(input())
    D = list(map(int,input().split()))
    mod = 998244353

    from collections import Counter

    node_cnt = Counter(D)
    if D[0]!=0 or node_cnt[0]!=1:
        print(0);exit()
    
    ans = 1
    max_depth = max(D)

    for i in range(max_depth):
        # if i not in node_cnt:
            # print(0);exit()
        # else:
        ans*=pow(node_cnt[i],node_cnt[i+1],mod)
    print(ans)

if __name__=='__main__':
    main()