Taipei to Sanya, layover in Xiamen

US citizen, traveling from Taipei to Sanya with a several hour layover in Xiamen. We will not leave the airport. We will return to Taipei in 3 days, also through Xiamen. (TPE XMN SYX) We are returning to Taipei, so we do not technically qualify to be visa free, as we go from country A to B and back to A.. Is this correct? If so, what do we need to do? Again, we are not leaving the airport in Xianan, and no visa is required for Hainan, from what I understand. Thanks.