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 <algorithm>
#include <vector>
#include <queue>
#include <iomanip>
#include <cmath>
#include <map>
using namespace std;
using ll = long long;
constexpr ll MOD = 998244353;

ll mod_pow(ll x, ll e) {
  ll ret = 1;
  while(e) {
    if(e&1) ret = ret * x % MOD;
    x = x * x % MOD;
    e >>= 1;
  }
  return ret;
}

int main() {
  ll n; cin >> n;
  ll a;
  vector<ll> cnt(n+1);
  for(int i = 0; i < n; ++i) {
    cin >> a;
    ++cnt[a];
  }
  if(cnt[0] != 1) {
    cout << 0 << endl;
    return 0;
  }
  ll ans = 1;
  for(int i = 1; i < n; ++i) {
    if(cnt[i+1] > 0 && cnt[i] == 0) {
      cout << 0 << endl;
      return 0;
    }
    ans = ans * mod_pow(cnt[i], cnt[i+1]) % MOD;
  }
  cout << ans << endl;
  return 0;
}