
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 come to the club and each gives his hat to the doorman. Each hat fits exactly to its owner, 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 doorman tries to put on his head the first available hat. If it fits, the gentleman leaves with the 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 tells the gentleman, "Sir, today the hat does not suit you," and the gentleman goes home without a hat. Find the probability that next Friday the doorman will not have any hats left.

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