Can a president run again if they served 4 years but try again and lose?

yes. two term limit, does not have to be back to back. Technically it's 10 years. A vice president that takes office in the event the president can no longer run the country can run for office an additional 2 terms UNLESS he took power prior to the 2nd complete year of the prior president. If the elected president can no longer remain in power and the vice president takes over the office before the second year into that term he can only run for office himself once.