I'd argue that AI's do not "think" they run mathematical sequences at a complexity that makes them capable of interacting with humans. However they are bound by logic, and I believe that is defines actual intelligence. Very few day to day decisions we make are based solely on logic. Most are influenced by more instinctual urges even at a base level. Urges that are bred into living things as a way to keep them alive.
So I'd maintain I think therefore I am; but I think illogically therefore I exist.\
edit: also above I should' have asked what is the actual thought process of becoming self aware? What is that transition between what what and what now is.