There is no service on OWE from PPT-AKL so you must return to SCL (this becomes PPT-SCL-AKL but only uses 1 extra segment despite stopping in IPC). Similarly AKL-PER is really AKL-SYD-PER, also there is no PER-BKK service at the moment, only PER-SIN so you must go PER-HKG-BKK or PER-SIN-BKK. You can download a Oneworld timetable to your PC and plan out your itinerary. The best one is at http://www.cathaypacific.com/intl/pl...le/0,,,00.html
(be sure to download the 3rd one, which include Oneworld flights). Use the exact dates for your trip, as flights vary by day of week and also by month.
Your final routing will be something like AMS-LHR-EZE-SCL-PUQ-SCL-IPC-PPT-SCL-AKL-SYD-PER-HKG-BKK-LHR-AMS (requires purchasing 2 extra segments in South America, which is allowed, and you are still under the 20 segment limit so you could do some more flights in Asia, maybe HKG-DPS-HKG or HKG-NRT-HKG or in Australia or Europe).