I realize this isn't the answer you want, but my recommendation is to ask your son what he wants. Where does he see himself as an adult? What makes you feel the Ivy League schools are the only ones that can provide the education for it? Are you going to be devastated if he doesn't get in? Why?
My nephew attended a city of Chicago public magnet school, and got in to M.I.T. but decided to attend U.of FL. instead. He may get his Master's or PhD from M.I.T. later, but he decided it wasn't the best fit right now.
Colleges aren't all looking for kids with "homogeneous private school backgrounds." A student who has good grades and shows a passion for something is likely to be looked at better than the kid who joined a dozen organizations just to "fill up his application resume." I'd like to suggest that you read this recent discussion, and consider your son's desires, too: community.greatschools.net/groups/11551/discussion/124824
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