A gas turbine engine includes a first spool associated with a primary combustor, a second spool associated with a secondary combustor, and a third spool, each spool including a compressor and a turbine mounted to a shaft. A transmission and accessory gearing are enclosed within an housing of an accessory gearbox. The transmission rotationally couples the third spool to the second spool and accessory gearing. A method of operating the gas turbine engine includes supplying a first fuel flow rate to the primary combustor and supplying a second fuel flow rate to the secondary combustor within an intermediate speed range of the gas turbine engine.