2 Round Monty Hall

I heard this version: It is the Monty Hall (MH) problem but in 2 rounds. A and B decide on their strategy ahead of time. Then A enters an instance of MH to choose the door with the car behind it, and if he succeeds, B enters the same instance of MH to choose the keys to the car. What is the maximum probability of success? Notes: There is no communication between A and B once the game begins. It is not success if A finds the key and B the car. This puzzle has apparently hit the blog world.