Travelling through South America Aug-Dec 2025, best route for efficiency and climate?

Hi all, hoping to do a solo backpacking trip next year from early August to Mid December.

It’s the only time frame I can realistically go and I just wanted some advice about how to best travel through the continent whilst hitting the best climate of each place.

I want to visit (in no particular order and with some places I’d love to see) Brazil (Rio), Argentina (watch a Pumas rugby game), Patagonia, Chile, Peru (Inca trail, Machu Picchu) and Colombia.

I’m not sure the best way to hit all the above but if anyone could advise the best route and best weather it’d be really appreciated

(Flying from LDN GW)

If there’s anything else which is a must do please let me know and I understand you can’t control the weather so I’m happy to take on realistic criticism

Thanks very much