General aviation pilot reported a near miss with another aircraft while on final approach at a non-towered airport. The pilot did not see the aircraft and the other pilot maneuvered to avoid a conflict.

Narrative

I was flying from ZZZ to AVQ to leave my plane for maintenance. There were several planes in the pattern at AVQ and around the airport - it was busy. I entered a downwind on the 45. Two planes were ahead of me. I was following Aircraft Z. Aircraft A was behind me and another plane behind him. All entering the pattern on the downwind. Aircraft Z turned final and announced #2 for landing. I was watching him and announced #3 for landing. Aircraft A announced #4. I had to extend downwind for spacing; as did Aircraft A. The plane behind him had to break off because continued downwind extensions would take him into Pinal Airpark (MZJ) airspace. As I turned final; Aircraft Y practicing a straight-in IFR approach announce that I had cut in under her and she was going around. Review of internet data showed that Aircraft Y was about 500 feet above me. I had been focused on maintaining visual contact with Aircraft Z ahead of me so as to land a close behind him as I safely could; making room for planes behind me. He exited the runway shortly before I touched down. My landing was uneventful; and the IFR plane entered the pattern from the upwind for a touch and go before returning to ZZZ1. I had heard the IFR plane announce an approach; but had lost awareness of where she was. I've heard that combining straight in and pattern work at uncontrolled airports is dangerous; but now have first-hand experience to verify.