US Passport holder - Do I qualify for 72/144 free transit

Hi there - I'm planning a trip and would like to know if I qualify for visa-free transit. My itinerary involves flying overnight from Houston (IAH) to Amsterdam (AMS), then traveling to Beijing (PEK). I intend to stay either 72 or 144 hours in Beijing before departing for Austin (AUS) with a layover in Frankfurt (FRA). Could you please let me know if I meet the requirements for visa-free transit?

IAH > AMS > PEK > FRA > AUS

Thank you,