Best way to spend 3 weeks in South America

My wife and I are looking to travel to South America for 21 days from Miami in October and am wondering if anyone has a way that they’ve done this sort of trip and loved.

We would like to visit Lima, Machu Picchu, go to Puerto Maldonado, Buenos Aires, Santiago (I have family there but this is the least important leg of the trip as I can go at any time) and Rio De Janiero.

However we are open to any and all suggestions as long as they include Machu Picchu.

I appreciate all of the advice and experiences you can share. Thanks!