Dan72,

Based on your description it seems TAM's flight is a continuation of an international service, thus you're expected in GRU to do an international-international connection: Just clear security to access the international gates, stay within the international sterile area to board the TAM flight as an international passenger and clear immigration and customs in Rio only. This flight arrives/departs within GRU international area and arrives in Rio at the international terminal so that all passengers get screened and cleared there.

If you start going down stairs you are going the wrong direction: Going down the stairs are for passengers clearing immigration/customs in GRU.

Once you've cleared security at GRU and access the international area, if you want to visit the lounges, head upstairs to the international lounges area. You will find TAM and United lounges within the T1 international sterile area next to each other. If you get lost, stairs upstairs are before all the gates and duty free shops.

Enjoy...