What is the most logical next event?

How to take care of a stray cat
Identify a lost pet cat.
Pet cats do not behave in the same way feral ones do, although if a cat has been lost for some time or is frightened, it may behave in a way that appears feral. If the cat approaches you and does not appear to fear humans, it is likely a lost pet.
A collar is an easy way to identify a lost pet cat. Pet cats may appear less clean than feral cats because they are unaccustomed to living outside and needing to clean themselves more frequently.