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I think that the $12k will be needed entirely for travel, visas and other administrative costs. Some of the top schools (Harvard, Yale) may give you the entire COA, but most do not. They do not pay for travel or clothing or extras you may want, like a new computer every year (often they do pay for one freshman year). We don’t know where your family is getting the $12k from, but if they earn more but pay for several generations of children or grandparents, own multiple homes (even if modest or they are renting them out), they may find you can afford to pay more than $12/yr. Most schools in the US do not give allowances for the cultures of other countries where families support relatives and use money that could go to college tuition.
You have a plan. You don’t want our opinions on picking other schools. Some of the schools we are suggesting DO award a few full ride scholarships. My daughter didn’t get a full ride from her school but did patch together a way to get the equivalent of a full ride through TEN forms of financial aid (merit, athletic, financial need). Some of those aren’t available to you (federal loans, state aid) but you need to figure out your way. She didn’t ‘live large’ and never went on spring break or study abroad, but she got a good education in engineering. It wasn’t from an Ivy or MIT
If your only way is to apply to the 20 top schools and cross your fingers, good luck. I think there are other schools that you could make work by you earning more (maybe a year off to just work, maybe online for a year or two).