An aircraft that uses Turbofan or Turboprop engines would be taking the compressed air from 3rd stage and 6th stage (varies) and supplying it to the Pressurisation and Air Conditioning Kit (PACK). This air is cooled, filtered, and then supplied to the Fdeck, FWD and AFT cargo zones for temperature control and breathing. If my understanding is correct, PACK is a kind of compressor itself which cools the bleed air for circulation purposes.
Because the engine has got LP and HP stages (i.e compressors), why can't the engine send the LP stage air into PACK just for filtering? Because LP air is surely cool air which I understand still requires filtering. Without knowing further , it surely saves a lot of effort for installing more compressors and therefore, creating a more complex pneumatic system. Has this not been done or is there something I am missing?