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

As each gentleman leaves, the porter tries to place the first hat he grabs on the gentleman’s head. If it fits, the gentleman leaves with that hat. If it does not fit, the porter tries another random hat from the remaining ones. If none of the remaining hats fit, the porter says, "Sir, the hat does not suit you today," and the gentleman has to leave with his head uncovered. Find the probability that next Friday, the porter will have no hats left at the end of the evening.

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