Hong Kong to Taiwan to Beijing to Taiwan to USA, 72 hours free transit?

I am US citizen travelling from USA to Hong Kong (7 day stay) and then from Hong Kong to Taiwan (6 day stay) can I go to Beijing using 72 hours and then go back to Taiwan and to USA, or do I need to go from Beijing to USA? I already have my return ticket from Taiwan to USA and wanted to get at least 48 hours in Beijing if possible.
USA - Hong Kong (7 days)
Hong Kong - Taiwan (6 days)
Taiwan - Beijing ( 2 days)
Beijing - Taiwan - USA

Thank you.