The following steps outline how Switch3 chooses a new RP:

Step 1. A new connection between Switch1 and Switch3 has been set up.

Step 2. Switch3 receives a better BPDU from Switch1. It keeps the new port in blocking state.

Step 3. Switch3 transitions the current RP to Alternate port (discarding state).

Step 4. Switch3 sends an agreement BPDU to Switch1.

Step 5. Switch3 transitions the new RP to forwarding state.

Step 6. Switch1 receives the agreement BPDU and transitions its port to forwarding state as well.