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 Kotlin (1.0.0) (judge verdict: Wrong Answer)

val MOD = 998244353

fun main(args : Array<String>) {
    val n = readLine()!!.toInt()
    val dd = readLine()!!.split(" ").map{it.toInt()}

    if (dd[0] != 0) {
        println(0)
        return
    }

    val groups = dd.groupBy { it }
    val max = groups.keys.max()!!

    var sum = 1L
    for (i in max downTo 1) {
        val cnt = groups[i]?.count() ?: 0
        val nextCnt = groups[i-1]?.count() ?: 0
        if (cnt == 0 || nextCnt == 0) {
            sum = 1L
            continue
        }

        sum *= modPow(nextCnt, cnt)
    }

    println(sum % MOD)
}

fun modPow(base: Int, pow: Int): Int {
    var ans = 1
    repeat(pow) {
        ans *= base
        ans %= MOD
    }
    return ans
}