A method of distributing power within a gas turbine engine is disclosed. In various embodiments, the method includes driving a high pressure turbine having a first stage and a second stage with an exhaust stream from a combustor, the first stage connected to a high pressure turbine first stage spool and the second stage connected to a high pressure turbine second stage spool; driving a high pressure compressor connected to a high pressure compressor spool via a differential system, the differential system having a first stage input gear connected to the high pressure turbine first stage spool, a second stage input gear connected to the high pressure turbine second stage spool and an output gear assembly connected to the high pressure compressor spool; and selectively applying an auxiliary input power into at least one of the high pressure compressor spool and the high pressure turbine.