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)

#%%
import sys
def input():
    return sys.stdin.readline().rstrip()

import math
def func(i, j, d):
    if i <= 1:
        return 1
    e = j
    b = int(math.log2(d)/math.log2(i))
    ret = 1
    while e > b:
        ret = ret*(i**b%d)
        e = e - b
    else:
        if e > 0:
            ret = ret*(i**e%d)
    return ret


def main():
    N = int(input())
    Ds = [int(d) for d in input().split()]

    Ds.sort()
    if Ds.count(0) != 1:
        print(0)
        return

    i = 1
    l = 1
    r = 1
    ans = 1
    divider = 998244353

    while i < N:
        cnt = 0
        while i < N and Ds[i] == l:
            cnt = cnt + 1
            i = i + 1
        if cnt == 0:
            print(0)
            return

        ans = ans*func(r, cnt, divider)%divider
        r = cnt
        l = l + 1
    
    print(ans)

# %%
if __name__ == '__main__':
    main()

# %%
# from atcoder_test import doTest
# doTest("nikkei2019-2-qual","b",main)