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)

def powmod(n: int, p: int, M: int) -> int:
    ans = 1
    for _ in range(p):
        ans *= n
        ans %= M
    return ans

def solve():
    N = int(input())

    distances = list(map(int, input().split()))

    node = [0] * N

    for distance in distances:
        node[distance] += 1

    ans = 1
    MOD = 998244353

    index = 0
    for i, n in enumerate(reversed(node)):
        if n != 0:
            index = len(node)-1 - i
            break
    #print(f'{index=}')
    for n in node[:index]:
        if n == 0:
            print(0)
            return
            

    if node[0] != 1:
        print(0)
    else:
        previous = node[1]
        for i in range(2,N):
            now = node[i]
            ans = ans * powmod(previous, now, MOD) % MOD
            previous = now
        print(ans)
    
solve()