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 JavaScript (node.js v5.12) (judge verdict: Runtime Error)

function main(arg) {
    var arg2 = arg.split("\n")[1].split(/\s/g); //元データ
    var kazoe = []; //各数字の個数の配列

	for(i=0;i<arg.split("\n")[0];i++){
		//もし存在しなければ作る
		if(!kazoe[arg2[i]]){
			kazoe[arg2[i]] = 0;
		}
		kazoe[arg2[i]] += 1;
	}
	//答えは(1の個数)^(2の個数)*(2の個数)^(3の個数)*(3の個数)^(4の個数)*…*(N-1の個数)^(Nの個数)
	var ans = 1;
	for(j=2;j<kazoe.length;j++){
	    ans = ans * (kazoe[j-1] ** kazoe[j]);
	}
	//もし間に歯抜けがあればfalse
	for(k=0;k<kazoe.length;k++){
	    if(!kazoe[k]){
	       ans = 0;
	       break;
	    }
	}
    console.log(ans);
}
main(require('fs').readFileSync('/dev/stdin', 'utf8'));