Scenario

Sue, a first-year law student, puts a poster up in her apartment building offering a reward of $1000 to anyone who finds and returns her cats Tabby and Toby. She includes a clear photo of her cats in the poster. She does not hear anything for a while but eventually Jane, another lawstudent in the building, knocks on Sue's door and produces the cats. Although Sue does not like Jane at all, she thanks her and starts to close the door. 'What about my reward' Jane says loudly. 'Oh, that was just my way of saying I love my cats, you don't think anyone would really pay so much just to get their pet returned?' 'Well, you made an offer', said Jane. 'Well, even if I did, you did not accept it,' said Sue.

Does Sue have to pay Jane the $1000? Identify the relevant legal principles and apply them to this situation.