Race - Should I Put Both or One??

I’m half black and half Asian (Taiwanese) with a last name that does not give away my Asian heritage.
Should I just put black on my college application? Or should I check both black and Asian?
I heard that being multicultural is better than being in just one race group, but I also heard that it’s better to not include Asian in the application at all. So I’m not sure what to put ???

I don’t know if it matters but my GPA is a 4.29 weighted and 3.73 unweighted and I got a 2020 on the SAT and I did not take the ACT, and my class rank is 25 out of 194.

I’m applying to mostly Tier 0 and Tier 1 schools but my top choices are Claremont McKenna and Tulane University (both out-of-state) and I’m hoping to not pay any more 10K (my household income is about 80K) so I’m hoping for merit scholarships and financial aid.

If you consider yourself half black and half asian, then put that down.

Whatever you consider yourself to be. When people ask you your race: Do you say you’re asian, do you say you’re black, or do you say you’re half?
On standardized testing, what do you identify yourself as?

That answer is the one you put on your college app. Race doesn’t matter for financial aid and Claremont McKenna I’m pretty sure doesn’t give any merit aid.