
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 his hat to the cloakroom attendant. Each hat exactly fits its owner, but there are no two hats of the same size. The gentlemen leave one at a time in a random order.

As each gentleman leaves, the cloakroom attendant tries to put the first available hat on his head. If it fits, the gentleman leaves with that hat. If it is too small, the cloakroom attendant tries the next random hat from the remaining ones. If all the remaining hats are too small, the attendant says to the gentleman: "Sir, today, no hat suits you," and the gentleman goes home with an uncovered head. Find the probability that the cloakroom attendant will have no hats left next Friday.

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