A note worthy point is that, Paxos actually expects proposers to drop their value and carry-forward someone-else's value to complete the consensus. This is typically not an intended behavior from client's point of view, but proposers role is to form the consensus, not push their value.
Read phase is basically understanding if there is any value that could've been chosen when F nodes are unavailable.
Determine phase is to choose to carry-forward any value (if-any) from the Read phase or use the new value from the client.
Write phase is to actually ask majority of the acceptors to save the value selected in the Determine phase.