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 = 998244353L

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

    if (dd[0].key != 0 || dd[0].value.size != 1) {
        println(0)
        return
    }

    var ans = 1L

    for (i in 1 until dd.size) {
        if (dd[i].key != i) {
            println(0)
            return
        }

        ans *= modPow(dd[i-1].value.size.toLong()%MOD, dd[i].value.size.toLong()%MOD, MOD)
        ans %= MOD
    }

    println(ans)
}

fun modPow(base: Long, p: Long, m: Long): Long {
    return when {
        p == 0L -> 1L
        p == 1L -> base
        p % 2L == 0L -> modPow(base, p / 2L, m).let { it*it%m }
        else -> base * modPow(base, p - 1L, m) % m
    }
}