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 Common Lisp (SBCL 1.1.14) (judge verdict: Wrong Answer)

(defun f (n m)
  (expt n m))

(let* ((n (read))
       (arr (make-array n :element-type 'fixnum :initial-element 0))
       (ans 1))
  (loop :repeat n :do(incf (aref arr (read))))
  (loop :for k :from 0 :upto (- n 2) :do(setf ans (* ans (mod (f (aref arr k) (aref arr (1+ k))) 998244353))))
  (princ (mod ans 998244353)))