Problem:
Score : 500 points

Problem Statement

Let us denote by f(x, m) the remainder of the Euclidean division of x by m.

Let A be the sequence that is defined by the initial value A_1=X and the recurrence relation A_{n+1} = f(A_n^2, M).
Find \displaystyle{\sum_{i=1}^N A_i}.

Constraints

1 \leq N \leq 10^{10}

0 \leq X < M \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X M

Output

Print \displaystyle{\sum_{i=1}^N A_i}.

Sample Input 1

6 2 1001

Sample Output 1

1369

The sequence A begins 2,4,16,256,471,620,\ldots Therefore, the answer is 2+4+16+256+471+620=1369.

Sample Input 2

1000 2 16

Sample Output 2

6

The sequence A begins 2,4,0,0,\ldots Therefore, the answer is 6.

Sample Input 3

10000000000 10 99959

Sample Output 3

492443256176507

Solution in Kotlin (1.3.71) (judge verdict: Accepted)

import java.io.BufferedReader
import java.io.InputStream
import java.io.InputStreamReader
import java.io.PrintWriter
import java.util.*

fun PrintWriter.solve(sc: FastScanner) {
    val n = sc.nextLong()
    val x = sc.nextInt()
    val m = sc.nextInt()
    var v = x.toLong()
    val sum = Array(m + 5) { v }
    val a = Array(m + 5) { v }
    val map = mutableMapOf<Long, Int>()
    map[v] = 0
    val size = if (n < m + 5) n.toInt() else m + 5
    for (i in 1 until size) {
        v = (v * v) % m
        a[i] = v
        sum[i] = sum[i - 1] + v
        if (!map.containsKey(v)) {
            map[v] = i
        } else {
            val j = map[v]!!
            val p = i - j
            val psum = sum[i - 1] - if (j > 0) sum[j - 1] else 0
            var ans = if (j > 0) sum[j - 1] else 0
            ans += psum * ((n - j) / p)
            val rem = ((n - j) % p).toInt()
            for (k in j until j + rem) {
                ans += a[k]
            }
            println(ans)
            return
        }
    }
    println(sum[n.toInt() - 1])
}

fun main() {
    val writer = PrintWriter(System.out, false)
    writer.solve(FastScanner(System.`in`))
    writer.flush()
}

class FastScanner(s: InputStream) {
    private var st = StringTokenizer("")
    private val br = BufferedReader(InputStreamReader(s))

    fun next(): String {
        while (!st.hasMoreTokens()) st = StringTokenizer(br.readLine())

        return st.nextToken()
    }

    fun nextInt() = next().toInt()
    fun nextLong() = next().toLong()
    fun nextLine() = br.readLine()
    fun nextDouble() = next().toDouble()
    fun ready() = br.ready()
}