Problem p02550 (C++) — buggy submission, verdict: Time Limit Exceeded

// g++ -std=c++11 a.cpp
#include <algorithm>
#include <assert.h>
#include <bitset>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <iomanip>
#include <iostream>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <typeinfo>
#include <unordered_map>
#include <utility>
#include <vector>
#define loop(i, a, b) for (long long i = a; i < b; i++)
#define rep(i, a) loop(i, 0, a)
#define FOR(i, a) for (auto i : a)
#define pb push_back
#define all(in) in.begin(), in.end()
#define shosu(x) fixed << setprecision(x)
#define show1d(v)                                                              \\
  {                                                                            \\
    rep(_, v.size()) cout << " " << v[_];                                      \\
    cout << endl;                                                              \\
  }
#define show2d(v)                                                              \\
  { rep(__, v.size()) show1d(v[__]); }
using namespace std;
// kaewasuretyuui
typedef long long ll;
#define int ll
typedef int Def;
typedef pair<Def, Def> pii;
typedef vector<Def> vi;
typedef vector<vi> vvi;
typedef vector<pii> vp;
typedef vector<vp> vvp;
typedef vector<string> vs;
typedef vector<double> vd;
typedef vector<vd> vvd;
typedef pair<Def, pii> pip;
typedef vector<pip> vip;
#define mt make_tuple
typedef tuple<int, int, int> tp;
typedef vector<tp> vt;
typedef vector<vt> vvt;
template <typename A, typename B> bool cmin(A &a, const B &b) {
  return a > b ? (a = b, true) : false;
}
template <typename A, typename B> bool cmax(A &a, const B &b) {
  return a < b ? (a = b, true) : false;
}
const double PI = acos(-1);
const double EPS = 1e-9;
Def inf = sizeof(Def) == sizeof(long long) ? 2e18 : 1e9 + 10;
int dx[] = {-1, 0, 1, 0};
int dy[] = {0, 1, 0, -1};
#define yes cout << "Yes\\n"
#define no cout << "No\\n"

signed main() {
  ios::sync_with_stdio(false);
  cin.tie(0);

  int n, x, m;
  cin >> n >> x >> m;
  int out = x;
  n--;

  vi used(m), dp(m);
  used[x] = 1;
  dp[x] = x;
  bool h = false;
  while (n) {
    int nx = x * x % m;
    out += nx;
    n--;
    if (!h && used[nx]) {
      out += (out - dp[nx]) * (n / (used[x] + 1 - used[nx]));
      n %= used[x] + 1 - used[nx];
      h = true;
    }
    dp[nx] = dp[x] + nx;
    used[nx] = used[x] + 1;
    x = nx;
  }

  cout << out << endl;
}


Error: TLE

Fixed submission:

// g++ -std=c++11 a.cpp
#include <algorithm>
#include <assert.h>
#include <bitset>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <iomanip>
#include <iostream>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <typeinfo>
#include <unordered_map>
#include <utility>
#include <vector>
#define loop(i, a, b) for (long long i = a; i < b; i++)
#define rep(i, a) loop(i, 0, a)
#define FOR(i, a) for (auto i : a)
#define pb push_back
#define all(in) in.begin(), in.end()
#define shosu(x) fixed << setprecision(x)
#define show1d(v)                                                              \\
  {                                                                            \\
    rep(_, v.size()) cout << " " << v[_];                                      \\
    cout << endl;                                                              \\
  }
#define show2d(v)                                                              \\
  { rep(__, v.size()) show1d(v[__]); }
using namespace std;
// kaewasuretyuui
typedef long long ll;
#define int ll
typedef int Def;
typedef pair<Def, Def> pii;
typedef vector<Def> vi;
typedef vector<vi> vvi;
typedef vector<pii> vp;
typedef vector<vp> vvp;
typedef vector<string> vs;
typedef vector<double> vd;
typedef vector<vd> vvd;
typedef pair<Def, pii> pip;
typedef vector<pip> vip;
#define mt make_tuple
typedef tuple<int, int, int> tp;
typedef vector<tp> vt;
typedef vector<vt> vvt;
template <typename A, typename B> bool cmin(A &a, const B &b) {
  return a > b ? (a = b, true) : false;
}
template <typename A, typename B> bool cmax(A &a, const B &b) {
  return a < b ? (a = b, true) : false;
}
const double PI = acos(-1);
const double EPS = 1e-9;
Def inf = sizeof(Def) == sizeof(long long) ? 2e18 : 1e9 + 10;
int dx[] = {-1, 0, 1, 0};
int dy[] = {0, 1, 0, -1};
#define yes cout << "Yes\\n"
#define no cout << "No\\n"

signed main() {
  ios::sync_with_stdio(false);
  cin.tie(0);

  int n, x, m;
  cin >> n >> x >> m;
  int out = x;
  n--;

  vi used(m), dp(m);
  used[x] = 1;
  dp[x] = x;
  bool h = false;
  while (n) {
    int nx = x * x % m;
    if (!h && used[nx]) {
      out += (out - dp[nx] + nx) * (n / (used[x] + 1 - used[nx]));
      n %= used[x] + 1 - used[nx];
      h = true;
    }
    used[nx] = used[x] + 1;
    dp[nx] = dp[x] + nx;
    out += nx;
    x = nx;
    n--;
  }

  cout << out << endl;
}