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: Accepted)

<?php
$n = (int)trim(fgets(STDIN));

$input = explode(' ', trim(fgets(STDIN)));
if ((int)$input[0] !== 0) {
	echo 0;
	exit();
}

$list = [];
for ($i=0; $i < $n; $i++) { 
	$list[$i] = 0;
}
foreach ($input as $v) {
	$list[(int)$v] += 1;
}
if ($list[0] !== 1) {
	echo 0;
	exit();
}
// var_dump($list);
$cnt = 0;
$mod = 998244353;
foreach ($list as $k => $v) {
	if ($k == 0) {
		$prev = 1;
		$cnt = 1;
	}else{
		if ($v != 0 && $prev == 0) {
			echo 0;
			exit();
		}
		$cnt = ($cnt * binPow($prev, $v, $mod)) % $mod;
		$prev = $v;
	}
}
echo $cnt;


function binPow($x, $n, $mod){
	$ans = 1;
	$bin = decbin($n);
	$cur_pos = strlen($bin) -1;
	while ($cur_pos >= 0) {
		if ($bin[$cur_pos] == '1') {
			$ans = ($ans * $x) % $mod;
		}
		$x = ($x * $x) % $mod;
		--$cur_pos;
	}
	return $ans;
}