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 (Clang 3.8.0) (judge verdict: Accepted)

#include<stdio.h>
#include<string.h>
#include<iostream>
#include<algorithm>
#include<string.h>
#include<math.h>
#define ll long long
using namespace std;
ll b[100005];
int main()
{
    int n;
    scanf("%d",&n);
    
    ll a[100005],i;
    
    for(i=0; i<n; i++)
        scanf("%lld",&a[i]);
    memset(b,0,sizeof(b));
    if(n>=1e5)
        a[-2]=0;
    if(a[0]!=0)
            printf("0\n");
    else
    {
        ll ans=1;
        for(i=0;i<n;i++)
            b[a[i]]++;
        for(i=1;i<n;i++)
        {
            ans*=b[a[i]-1];
            ans%=998244353;
        }

        printf("%lld\n",ans);
    }
    return 0;
}