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 (2.4.0) (judge verdict: Wrong Answer)

from fractions import gcd
import os
import math
import sys
from math import factorial
input = sys.stdin.readline


if os.getenv("LOCAL"):
    sys.stdin = open("_in.txt", "r")

sys.setrecursionlimit(2147483647)
INF = float("inf")
IINF = 10 ** 18
MOD = 998244353

g1 = [1, 1]  # 元テーブル
g2 = [1, 1]  # 逆元テーブル
inverse = [0, 1]  # 逆元テーブル計算用テーブル


def cmb(n, r, mod):
    return factorial(n) / factorial(r) / factorial(n - r)
    # if (r < 0 or r > n):
    #     return 0
    # r = min(r, n-r)
    # return g1[n] * g2[r] * g2[n-r] % mod


def main():
    N = int(input())
    d = list(map(int, input().split()))
    D = []
    d = sorted(d)

    for (k, v) in enumerate(d):
        D.append((v, k+1))

    distance = [0] * (max(d) + 1)

    for _d in D:
        distance[_d[0]] += 1

    if d[0] != 0:
        print(0)
        return

    if len(distance) != len([i for i in distance if i != 0]):
        print(0)
        return

    if distance[0] != 1:
        print(0)
        return

    ans = 1
    for i in range(0, len(distance)):
        if i == 0:
            ans = 1
            # print(ans)
        else:
            n = distance[i-1]
            r = distance[i]
            ans = ans * n ** r
            ans = ans % MOD
            # print(ans)

    print(ans % MOD)
    # print(*distance)
    # n = 2
    # r = 3
    # print(cmb(n+r-1, r-1, MOD))


if __name__ == '__main__':
    main()