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 (3.4.3) (judge verdict: Wrong Answer)

n=int(input())
d=list(map(int,input().split()))
mod=998244353
d.sort()

if n==1:
	if d[0]==0:
		print(1)
		exit()
	else:
		print(0)
		exit()
elif n==2:
	if d==[0,1]:
		print(1)
		exit()
	else:
		print(0)
		exit()
if d[0]!=0 or d[1]!=1:
	print(0)
	exit()
for i in range(n-1):
	if d[i+1]>d[i]+1:
		print(0)
		exit()
p=[]
cnt=1
for i in range(1,n-1):
	if d[i]==0:
		print(0)
		exit()
	if d[i]==d[i+1]:
		cnt+=1
	elif d[i+1]==d[i]+1:
		p.append(cnt)
		cnt=1
p.append(cnt)
ans=1
for i in range(len(p)-1):
	ans*=pow(p[i],p[i+1],mod)
	ans%=mod
print(ans)