Light aircraft pilot reported an NMAC with another light aircraft in the pattern at S50.

Narrative

I entered a 45 for a right downwind for runway 17 at Auburn and had traffic in sight in front of me and announced it and said I would follow. I was number 3 to land; and the number 2 aircraft extended downwind for the traffic on final and so did I. While I was on downwind I heard Aircraft Y announce he was joining the downwind from the cemetery. This would place him behind me but he did not announce he had me in sight. When I turned base I announce turning extended base number 3 still at that point since the number 1 plane had not landed yet. As I was on final I heard Aircraft Y announce he was turning base number 3. At this point the number 1 plane had landed so I assumed he said number 3 because of that. He had still not reported me in sight at that point. I assumed he was behind me following me if he said he was number 3; and I made another call announcing I was on final number 2. I hear Aircraft Y announce he was turning final and that's when another aircraft jumped on the radio to announce to me of that aircraft; and subsequently while looking to my right I see Aircraft Y just off my right wing turning final. I immediately took evasive action and made a quickly made a 360 to get back on final; but ended up going around. In the future I will maintain situational awareness of all planes in the pattern by always verifying other aircraft's radio calls visually; and confirm with other aircraft that they have me in sight if I believe they're supposed to/are following me.