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 Pascal (FPC 2.6.2) (judge verdict: Accepted)

const p=998244353;
type ary =array[1..100010] of int64;
     
var n,ijk,ans,D1,l,now,last:int64;
    D:ary;
    img:boolean;
    
function pawarmd(a,b:int64):int64;
 var store,ij:int64;
begin
 store:=1;
 for ij := 1 to b do store := store * a mod p;
 pawarmd:=store;
end;
    
procedure sort(p,q :integer; var a :ary);
  var i, j, k ,w:integer;
      
begin i := p; j := q; k:=a[(p+q) div 2];
  repeat
    while a[i] < k do inc(i);
    while k < a[j] do dec(j);
    if i <= j then begin
      w := a[i]; a[i] := a[j]; a[j] := w;
      inc(i);dec(j);
    end
  until i > j;
  if p < j then sort(p,j,a);
  if i < q then sort(i,q,a)
end;   
       
begin
 read(n);
 ans:=1;
 img:=false;
 read(D1);
 l:=1;
 now:=0;
 last:=1;
 
 if D1<>0 then img:=true
 else
  begin
   for ijk := 1 to n-1 do read(D[ijk]);
   sort(1,n-1,D);
   for ijk := 1 to n-1 do
    if D[ijk]=l then inc(now)
    else 
     if D[ijk]-l >=2 then
      begin
       img:=true;
       break;
      end
     else
      begin
       ans:=ans*pawarmd(last,now) mod p;
      
       inc(l);
       last := now;
       now:=1;
      end;
  end;
  
  if not(img) then ans:=ans*pawarmd(last,now) mod p;
 
if img then write(0) else write(ans);
end.