
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 the club, and each gives the doorman his hat. Every hat fits its owner perfectly, but no two hats are of the same size. The gentlemen leave one by one in a random order.

When seeing off the next gentleman, the club doorman tries to put the first available hat on his head. If it fits, the gentleman leaves with that hat. If it doesn't fit, the doorman tries the next random hat from the remaining ones. If all the remaining hats are too small, the doorman says to the unfortunate gentleman, "Sir, the hat doesn't suit you today," and the gentleman goes home bareheaded. Find the probability that the doorman will have no hats left the following Friday.

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