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
define('MOD', 998244353);
fscanf(STDIN, "%d", $N);
$D = explode(' ', trim(fgets(STDIN)));
$max = max($D);
$array = array_pad(array(), $max + 1, 0);
for ($i = 0; $i < $N; $i++) {
    $array[$D[$i]]++;
}
$ans = 1;
for ($i = 0; $i < $max; $i++) {
    for ($j = 0; $j < $array[$i + 1]; $j++) {
        $ans *= $array[$i];
        $ans %= MOD;
    }
}
if ($D[0] !== "0") {
    $ans = 0;
}
for ($i = 1; $i < $N; $i++) {
    if ($D[$i] === "0") {
        $ans = 0;
    }
}
printf("%d", $ans);