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)

p=998244353
def pow(x,m):
    if m==0:
        return 1
    if m==1:
        return x
    if m%2==0:
        return (pow(x,m//2)**2)%p
    else:
        return (x*(pow(x,(m-1)//2)**2)%p)%p
N = int(input())
D = list(map(int,input().split()))
D.insert(0,0)
C = {i:0 for i in range(N)}
for i in range(1,N+1):
    C[D[i]] += 1
if C[0]!=1:
    cnt = 0
else:
    C = sorted(list(C.items()),key=lambda x:x[0])
    ind = 0
    for i in range(N):
        if C[i]==0:
            ind = i
            break
    flag = 0
    if ind>0:
        for i in range(ind+1,N):
            if C[i]>0:
                flag = 1
                break
    if flag==1:
        cnt = 0
    else:
        cnt = 1
        for i in range(1,N):
            if C[i][1]==0:break
            c0 = C[i-1][1]
            c1 = C[i][1]
            cnt = (cnt*(pow(c0,c1)))%p
print(cnt)