Flights from Auckland to Santiago South America?

I'm currently planning a RTW and my first stop is a 4 week trip around New Zealand. My plan was to go around NZ, North and South island then go back to Auckland and fly on from Auckland to maybe Santiago, Chile ot Rio De Janeiro, Brazil. But i seem to not be able to find flights to either of these places from Auckland. Obviously i could go from Sydney, but considering i'm from Sydney and will be starting my trip from Sydney, it seems silly to back track to get to South America. Is there a way to get to South America from NZ thats cheaper then going via Sydney?