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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <cmath>

using namespace std;
static const int MAX = 100000;

int main(void) {
  long long int N, i, Ds[MAX], Es[MAX];
  long long int ans = 1;
  cin >> N;
  for (i=0;i<N;i++) {
    cin >> Ds[i];
    Es[Ds[i]] ++;
  }

  if (Ds[0] != 0 && Es[0] != 1) {
    cout << 0 << endl;
    return 0;
  }

  for (i=0;i<N-1;i++) {
    if (Es[i] == 0) break;
    ans *= pow(Es[i], Es[i+1]);
  }

  cout << (ans % 998244353) << endl;

}