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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include<iostream>
#include<vector>
#include<algorithm>
#include<string>
#include<queue>
#include<cmath>
#include<cstdio>
#include<tuple>
#include<bitset>
#include<map>

using namespace std;
#define int long long
#define rep(i,n) for(int i=0;i<n;++i)
#define rep1(i,n) for(int i=1;i<=n;++i)
#define ALL(x) x.begin(),x.end()
#define debug(output) cout<<#output<<"= "<<output<<endl

using P=pair<int,int>;
using lint=long long;
using ll=long long;
const lint inf=1e18+7;
const int MOD=1000000007;
signed main(){
  lint n,x,m;cin>>n>>x>>m;
  vector<lint> a(m+5,0);
  a[1]=x;
  lint c=0;
  map<lint,int> memo;
  memo[a[1]]=0;
  lint key=0;
  bool first=true;
  for(int i=2;i<=m+1;++i){
    a[i]=a[i-1]*a[i-1]%m;
    if(memo.find(a[i])==memo.end()&&first){
      memo[a[i]]=i;//first=false;
    }
    else{
      c=i-memo[a[i]];
      key=memo[a[i]];
      memo[a[i]]=i;
    }
  }
  if(n<=m){
    lint res=0;
    rep1(i,n)res+=a[i];
    cout<<res<<"\n";
    return 0;
  }
  lint sm=0;
  for(int i=key;i<key+c;++i){
    sm+=a[i];
  }
  lint res=0;
  for(int i=1;i<key;++i)res+=a[i];
  res+=sm*((n-key+1)/c);
  for(int i=key;i<key+(n-key+1)%c;++i)res+=a[i];
  cout<<res<<"\n";
  //debug(sm);
  return 0;
}