> cout << "Type 1 to exit or 0 to continue." << endl;
> cin >> exit;
Put these two lines inside your inner while loop, so you ask after each response.
> while (!correct)
Change to
while (!correct && !exit)
exit is also the name of a function in cstdlib, so you might want to change the name of your local variable to save a bit of weirdness later on.