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: Accepted)

fun main(args: Array<String>) {
    val n = readLine()!!.toInt()
    val ds = Array(1000000) { 0 }
    val dTmp = readIntegers()
    for (i in 1..n) {
        val d = dTmp[i - 1]
        ds[d]++

        if (i == 1 && d != 0) {
            println(0)
            return
        }

        if (i != 1 && d == 0) {
            println(0)
            return
        }
    }

    var ans = 1L
    val mod = 998244353L
    for (i in 1 until (1000000 - 1)) {
        var pow = 1L
        for (j in 1..ds[i + 1]) {
            pow = (pow * ds[i]) % mod
        }
        ans = (ans * pow) % mod
    }
    println(ans)
}

fun readStrings() = readLine()!!.split(" ")
fun readIntegers() = readStrings().map(String::toInt)