
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
Every Friday, ten gentlemen come to the club, and each one leaves his hat with the doorman. Each hat perfectly fits its owner, but there are no two hats of the same size. The gentlemen leave one by one in a random order.

As each gentleman is seen off, the club doorman tries to fit the first hat he finds on the gentleman's 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 all the remaining hats are too small, the doorman says to the unfortunate gentleman: "Sir, today no hat fits you", and the gentleman goes home with an uncovered head. Find the probability that next Friday the doorman will be left with no hats.

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