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 <complex>
#include <iostream>
#include <stack>
#include <unordered_map>
#include <unordered_set>
#include <vector>
#include <queue>
#include <algorithm>
#include <cmath>
#include <functional>
#include <iterator>
#include <tuple>
#include <set>
#include <map>

using namespace std;

bool myfunction (int i, int j) { return (i > j); }

template <typename T>
T modpow(T base, T exp, T modulus) {
  base %= modulus;
  T result = 1;
  while (exp > 0) {
    if (exp & 1) result = (result * base) % modulus;
    base = (base * base) % modulus;
    exp >>= 1;
  }
  return result % modulus;
}

typedef unsigned long long u64;

u64 multiplyModulo(u64 a, u64 b, u64 c)
{
  u64 result = 0;
  a %= c;
  b %= c;
  while (b) {
    if (b & 0x1) {
      result += a;
      result %= c;
    }
    b >>= 1;
    if (a < c - a) {
      a <<= 1;
    }
    else {
      a -= (c - a);
    }
  }
  return result % c;
}


int main() {
    int n;
    cin >> n;
    
    map<u64, u64> m;
    for (int i = 0; i < n; i++) {
        int val;
        cin >> val;
        m[val]++;
    }
    
    if (m.count(0) != 1) { cout << 0 << endl; return 0; }
    u64 res = 1;
    for (auto &p: m) {
        if (p.first > 1) {
            res = multiplyModulo(res, (modpow(m[p.first - 1], p.second, (u64)998244353)), (u64)998244353);
        }
    }
    cout << res << endl;
    return 0;
}