I don't know about California schools either, but I think some chapters would think it pretty cool to have an exchange student, assuming you like them and they like you, of course. Getting a bid as a senior can be tougher depending on the campus, but again, that might be where the uniqueness of your student status might work in your favor. I say go through rush and see what happens, worst case scenario is that you don't get a bid, but at least you'll have had the experience of rush and won't always wonder "what if" if you don't try... good luck!!!
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