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 <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <map>
#include <bitset>
#include <vector>
#include <queue>

using namespace std;

typedef long long ll;
#define FOR(i,a,b)   for(ll i = (a); i < (b); i++ )
#define REP(i, n)    FOR(i,0,n)
typedef pair< ll, ll >   cp2;
typedef pair< ll, cp2 >   cp3;
#define fi first
#define se second
#define sec  se.fi
#define thr  se.se
const ll mod = 998244353;



///////////////////////////////////////////////
//
//
///////////////////////////////////////////////

////////////////////////////////////////////////
////////////////////////////////////////////////

int N;
int D[112345];
int cnt[112345];
int num = 1;
ll ans = 1;

int main(){
    
    cin>>N;
    
    REP( i, N ) scanf("%d", D+i );
    
    if( D[0] == 0 ){
        REP( i, N ) cnt[D[i]]++;
        if( cnt[0] == 1 ){
            FOR( i, 1, N ){
                REP( j, cnt[i] ){
                    ans = (ans*cnt[i-1])%mod;
                }
            }
            cout<<ans<<endl;
        }
        else{
            puts("0");
        }
    }
    else{
        puts("0");
    }


    return 0;
}