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 <bits/stdc++.h>
using namespace std;

int main() {
  int N;
  cin >> N;
  vector<int> D(N);
  for (int i = 0; i < N; i++) cin >> D[i];

  if (D[0] != 0) {
    cout << 0 << endl;
    return 0;
  }
  
  for (int i = 1; i < N; i++) {
    if (D[i] == 0) {
      cout << 0 << endl;
      return 0;
    }
  }

  vector<int> dist(N);
  for (int i = 0; i < N; i++) dist[D[i]]++;

  long long ans = 1;
  int v = 1;
  for (int i = 1; i < N; i++) {
    long long tmp = (dist[i - 1] == 0 ? 0 : 1);
    for (int j = 0; j < dist[i]; j++) tmp *= dist[i - 1];
    ans = ans * tmp % 998244353;
    v += dist[i];
    if (v == N) break;
  }

  cout << ans << endl;
  return 0;
}