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;
typedef long long ll;
template<class T = int> using V = vector<T>;
template<class T = int> using VV = V< V<T> >;
#define REP(i,n) for(int (i)=0;(i)<(n);(i)++)
#define ALL(v) (v).begin(),(v).end()

template <typename T>
T pow(T x, T y) { //Ruby no x**y
    if(y == 0) return 1;
    if(y&1 == 0) return pow(x*x, y/2);
    return x*pow(x, y-1);
}

int main() {
	const int mod = 998244353;
	int N;
	cin >> N;
	V<> counts(N,0), dist(N);
	REP(i,N) {
		int a;
		cin >> a;
		dist[i] = a;
		counts[a]++;
	}

	// REP(i,N) cout << dist[i] << " \n"[i==N-1];
	// REP(i,N) cout << counts[i] << " \n"[i==N-1];
	if(dist[0] != 0 || counts[0] != 1) {
		cout << 0 << endl;
		return 0;
	}
	int ans = 1;

	for(int i=1; i<N; i++) {
		if(counts[i-1] > 1) {
			ans = ((ans%mod) * (pow(counts[i-1],counts[i]%mod)))%mod;
		}
		if(counts[i-1] == 0) break;
	}
	cout << ans << endl;
}