The one thing I'm noticing a lack of in your post is references to extracurriculars. Your academics and GMAT are great (if you want to try for a higher score, go for it, but a 720 is great). From a work perspective, if you are successful in finding that PE firm job, then make sure you really look for leadership opportunities. But to round your application, you'll need extracurriculars. And by that I don't mean you volunteer at a soup kitchen once a week. It has to drive impact and tell a cohesive story. Schools like Columbia really value social and civic responsibility and you should find something you're passionate about.
The other thing to keep in mind at a high level is how are you going to stand out? Extracurriculars are one way, but the story needs to pop. IB to PE to MBA is a strong pedigree story but what's going to make them interested in YOU specifically? Perhaps it's really pursuing a hobby or perhaps it comes down to giving back. Whatever it is, schools like Columbia, Booth, MIT, Tuck, etc. get tons of applications from people just like you all the time. So find one or two things that you can really pursue for the next few years and have that build up your application's identity.
if you have any questions or want to chat about this in detail - don't hesitate to reach out!
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