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 stdout
printn = lambda x: stdout.write(x)
inn = lambda : int(input())
inl   = lambda: list(map(int, input().split()))
inm   = lambda:      map(int, input().split())
DBG = True # and False
BIG = 999999999
R = 998244353

def ddprint(x):
  if DBG:
    print(x)

n = inn()
d = inl()
#d.insert(0,0)
if d[0]!=0:
    print(0)
    exit()
a = [0]*n
for i in range(n):
    a[d[i]] += 1
if a[0]>1:
    print(0)
    exit()

#ddprint(d)
#ddprint(a)

p = 1
t = -1
for i in range(1,n):
    if a[i]==0:
        t = i
        break
    p = (p * (a[i-1]**a[i])) % R
if t>=0:
  for i in range(t+1,n):
    if a[i]>0:
        print(0)
        exit()
print(p)