
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 gives the doorman their hat. Each hat is exactly the right size for its owner, and no two hats are 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 he grabs. If it fits, the gentleman leaves with this hat. If it's too small, the doorman tries another random hat from the remaining ones. If all the remaining hats are too small, the doorman tells the poor fellow, "Sir, today's hat does not suit you," and the gentleman goes home with his head uncovered. Find the probability that next Friday the doorman will be left with no hats.

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