The Revised Monty Hall Problem

Many people have probably heard of the Monty Hall problem. Here is a different version of it.

You are playing on a game show and have made it to the last round. You have a chance to win a car! The game show host shows you three doors. He claims that behind one of the doors is the car, and behind the other two are goats. The game show host asks you to pick a door and explains what will happen after you pick your door. Before your door is opened, the host will open one of the other two doors at random. If that door reveals the car, you automatically win it! If it turns out to reveal a goat, you will have the option to swap your chosen door and open the door you didn't choose in the beginning. If the door you open reveals the car, you win the car.

So here is what happens: You pick the door on the right. The host chooses to open the door in the middle. As it turns out, that door was holding a goat. Now you have the option to switch. Should you switch, stick, or does it not matter what you choose to do?

Which action would give you the highest chance of winning the car?

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