Pilot reported loss of separation during landing; requiring the other aircraft to take evasive action. The pilot continued their approach to landing.

Narrative

I'm don't know whether I violated a clearance or not; but am reporting this anyway because there is something to be learned from it.I was cleared to enter L midfield downwind for landing [Runway] XX at ZZZ then cleared to land #3. At the time I entered; there were already three C172's in the pattern. According to my own situational awareness; there was one either 'on the go' or crosswind behind me that I couldn't see and one that I could clearly see in front of me on downwind and one somewhere in front of that one on base or perhaps final.I followed the one I could see (who I took to be '#2'); taking care not to lose sight of him while trying to keep my speed up to stay in front of the faster C172 I presumed was behind me. As I continued my approach I heard the Tower issuing taxi instructions (I think to the ramp) which supported my belief that #1 had now landed.As I was on final; another aircraft reported to ATC that I was now in front of them; initial separation had been only approximately 250 ft.; and they were going around.I honestly don't know what happened here. I may well have identified the wrong C172 as '#2'; or maybe '4' was closer and faster and flying a tighter pattern and overtook me. Without a radar or ADS-B plot it may be unknowable.The take aways for me are:- even with an active ATC; entering into a pattern that already has three identical planes in it is not a good idea (especially in somewhat flat lighting) unless the tower has radar/ADS-B. I had already voluntarily right-360'd once because I wasn't comfortable where everybody was; and should have done it again.- with several planes in the pattern doing T&G's; the '#1' (and #2; etc) label changes quickly. If you can't see both 1 and 2; confirm with ATC that the designated #2 is the one you think it is.- I've been wanting to install ADSB-IN in my own plane and this confirms why. Might have helped; whatever the root cause was.Finally; this is in no way a bust on ATC. These guys have always been great; and many times freakin' amazing at how they work. My hat remains off to them. Far more likely on average that I (or maybe another pilot; I don't know) screwed up rather than them.