tortise and hare doesn't make any sense... When the data is not stored up in any structure, the time is enough for just purely comparison and increment count accordingly.

    int a = s%mod;
    int compare = s%mod;
    int count = 1;
    for (int i = 1; i < n; i++)
    {
        a = ((a%mod)*(p%mod)+q%mod)%mod;
        if (a != compare)
        {
            count++;
            compare = a;
        }
        else continue;
    }
    cout<<count;