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 PHP7 (7.0.15) (judge verdict: Wrong Answer)

<?php
$n=trim(fgets(STDIN));
$d=explode(" ",trim(fgets(STDIN)));
if($d[0]!=0){
  echo 0;
}else{
  if($n==1){
    echo 1;
    exit;
  }
  $c=array_count_values($d);
  if($c[0]!=1){
    echo 0;
  }else{
    $t=count($c);
    $i=1;
    $ans=1;
    while($i<$t){
      if(!isset($c[$i])){
        echo 0;
        exit;
      }
      (int)$ans*=(pow($c[$i-1],$c[$i])%998244353);
      $ans%=998244353;
      ++$i;
    }
    echo $ans;
  }
}