
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Each Friday, ten gentlemen visit a club and each gives the doorman their hat. Each hat fits its owner perfectly, and there are no hats of the same size. The gentlemen leave the club one by one in random order.

As each gentleman leaves, the doorman tries to fit the first available hat on his head. If it fits, the gentleman leaves with that hat. If it is too small, the doorman tries the next random hat from the remaining ones. If none of the remaining hats fit, the doorman says to the unfortunate gentleman: "Sir, today the hat does not suit you," and the gentleman goes home with his head uncovered. Find the probability that next Friday the doorman will have no hats left.

Present the answer in LaTex format: \boxed{Your answer}
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