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: Accepted)

#include <iostream>
#include <string>
#include <array>
#include <vector>
#include <queue>
#include <stack>
#include <list>
#include <set>
#include <map>
#include <algorithm>
#include <cmath>
#include <iomanip>

#define FOR(i, a, b) for(decltype(b) i = (a); i < (b); ++i)
#define REP(i, n) FOR(i, 0, n)

typedef long long ll;
typedef unsigned long long ull;
typedef unsigned int uint;

using namespace std;

#ifdef INPUT
#include <sstream>
stringstream input(R"(
7
0 3 2 1 2 2 5
)");
#define cin input
#endif

int main()
{
	int n;
	cin >> n;

	map<int, int> d;

	int x;
	cin >> x;

	REP (i, n - 1)
	{
		int x;
		cin >> x;
		++d[x];
	}

	if (x != 0 || d.begin()->first != 1)
	{
		cout << 0 << endl;
		return 0;
	}

	ll r = 1;
	d[0] = 1;
	REP (i, n - 2)
	{
		REP (j, (int)d[i + 1])
		{
			r *= d[i];
			r %= 998244353;
		}
	}

	cout << r << endl;

	return 0;
}