You are in a network of rooms laid out in a 3 by 3 grid (as shown in the diagram). Each room has a door connecting it to rooms adjacent to it. There are 12 doors in total and all are open at the start of the game. You are placed in one of the corner rooms and your aim is to visit each room once. When you leave a room, all the doors connected to that room are automatically closed and locked. You may finish your route at any convenient room.

In the middle of each room, there is a gold coin placed on a table. You must enter a room in order to pick up each coin. If you have successfully collected all nine gold coins, you will be magically teleported to freedom!

The diagram shows one possible winning route. Starting from the same room in the top left corner of the network, how many different routes are possible so as to visit each room once?

Now, that was far too easy! The real game is slightly more challenging. In the first room, next to the gold coin, are two dodecahedral dice, each numbered 1 to 12, and a small map of the network. Each door in the network has a number on it from 1 to 12 inclusive and the map shows the location of each numbered door. You roll the two dice until you have two distinct numbers; you may only roll again if the two numbers are the same. You will then hear the sound of two doors being locked closed; they are the doors that correspond to the two numbers on the dice.

What is the probability that you can still pick up all nine gold coins, even after the two doors have been locked?

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