

U.S. --> Taipei (Taiwan) --> Hangzhou --> U.S. I will fly nonstop from Los Angeles to Taipei.
Stay for a few days in Taipei. Then, fly nonstop from Taipei to Hangzhou. Again, stay for a few days (less than six) in Hangzhou. Finally, I will fly from Hangzhou back to Los Angeles. A --> China --> C with A being Taiwan and C the U.S. It seems like this should qualify for the 144-hour visa exemption, does it?
The only caveat is that on my return flight from Hangzhou to U.S. there is no direct nonstop flight. I will have a two and half hours layover in Taipei (Taiwan) with plane change and a different flight number (from the flight of Hangzhou to Taipei). Even though I'm returning to the U.S., will it be in any way construed differently, i.e. The C being Taiwan not U.S. which won't work since A and C are the same.
I called the airline and travel agent with no help. Chinese embassy in Los Angeles won't answer phone calls. They did reply to my email inquiry with a website link to the policy which does not answer the question of connecting flight.
Please help! Thank you!