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

#include <cstdio>
#include <climits>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <algorithm>
#include <string>
#include <cstdio>
#include <climits>
#include <iostream>
#include <algorithm>
#include <string>
#include <vector>
#include <utility>
#include <queue>
#include <cstring>
#define rep(i, n) for (int i = 0; i < int(n); i++)
using namespace std;
long long MOD = 998244353;
long long INF = 1000000000000000;
#define MAX_V 330

typedef long long ll;

ll spow(ll a, ll b) {
  if (b == 1) return a % MOD;
  if (b == 0) return 1;
  ll c = spow(a, b / 2);
  if (b % 2 == 0) return (c * c) % MOD;
  else return (a * (c * c % MOD)) % MOD;
}

int main(void) {
  ll n; cin >> n;
  ll d[100010], c[100010] = {}, ma = 0;
  rep(i, n) {
    cin >> d[i];
    ma = max(ma, d[i]);
    c[d[i]]++;
  }
  ll ans = 1;
  for (int i = 1; i <= ma; i++) {
    if (c[i - 1] == 0) {
      cout << 0 << endl;
      return 0;
    }
    ans *= spow(c[i - 1], c[i]);
    ans %= MOD;
  }
  cout << ans << endl;
  return 0;
}