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)

from sys import stdin

N = int(stdin.readline().rstrip())
D = [int(x) for x in stdin.readline().rstrip().split()]

if D[0] > 0:
    print(0)
    exit()
D.sort()

ans = 1
mod = 998244353
b = 1

temp = 1
for i in range(1, N):
    if D[i] == 0:
        print(0)
        exit()
    if D[i] - D[i - 1] > 1:
        print(0)
        exit()
    if D[i] != D[i - 1]:
        b = temp
        temp = 1
    else:
        temp += 1
    ans *= b
    ans %= mod

print(ans % mod)