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With any award seat availability, you must have some sort of flexibility when choosing your flight dates. Give yourself lots of time. Say anywhere from 8 months to departure. Be flexible about your departure point. That means use the available options of flying from LAX (3 flights a day) , SFO (2 flights a day), YVR (2 flights a day), in addition to combining with say, JL/AA via NRT.
Look out also for peak travel periods. The Chinese Lunar New Year, Spring Festivals etc are usually packed departing anywhere within Asia.
With careful planning, you would be able to get the flights that you want. You may use the asiamiles website in looking for possible award availability with CX, but it is not that accurate. Best to perhaps call AA and check your seats.
Be aware that CX has very very limited award availability that is shared amongst all its travel partners.