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 (7.3.0) (judge verdict: Wrong Answer)

n=int(input())
mod=998244353
A=list(map(int,input().split()))
import sys
mod=10**9+7 ; inf=float("inf")
from math import sqrt, ceil
from collections import deque, Counter, defaultdict #すべてのkeyが用意されてる defaultdict(int)で初期化
input=lambda: sys.stdin.readline().strip()
sys.setrecursionlimit(11451419)
from decimal import ROUND_HALF_UP,Decimal  #変換後の末尾桁を0や0.01で指定
  #Decimal((str(0.5)).quantize(Decimal('0'), rounding=ROUND_HALF_UP))
from functools import lru_cache
from bisect import bisect_left as bileft, bisect_right as biright
from fractions import Fraction as frac  #frac(a,b)で正確なa/b
#メモ化再帰defの冒頭に毎回 @lru_cache(maxsize=10**10)
#引数にlistはだめ
#######ここまでテンプレ#######
#ソート、"a"+"b"、再帰ならPython3の方がいい
#######ここから天ぷら########

C=Counter(A)
Q=[C[i] for i in range(max(A)+1)]
#print(Q)
ans=1
for i  in range(len(Q)-1):
    ans*= pow(Q[i],Q[i+1],mod)
    ans%=mod

print(ans)