well... as those princeton students are not here. I'd like to saything in my opininion.
Princeton is a very selective program if you don't have some direct recommendation from their professors' acquaintance. Princeton really like extremely all around students, and they strongly appreciate working experience. But most important thing, due to their policy of diversity, Princeton never directly recruits two students of similar background in their program. That is to say, you have to be the best one of your pool or you have zero chance.
I guess it will be of very little doubt that LZ can get at least first round interview. But to be frank, interview is nothing. All best luck.