
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
A department store received 10 suitcases and 10 separate keys in an envelope, with the warning that each key opens only one suitcase and that a matching key can be found for each suitcase.

A store worker, sighing at the task of matching keys to suitcases, imagined having to try up to 10 times for one suitcase, and up to 100 times for all 10 suitcases.

The saleswoman reassured that the number of attempts needed would not exceed \(10+9+8+\ldots+2+1=55\), while another colleague noted that it could be reduced to \(9+8+\ldots+1=45\) as each unmatched key naturally fits the last suitcase. He also suggested in the worst case, each key matching on the last attempt, it would reflect this sum. They speculated the average number of attempts might be around half the maximum, that is, 22.5.

Igor Fedorovich Akulich from Minsk questioned this estimation, reasoning the calculation might be more nuanced than the suggested 22.5 attempts.

**Task**: Determine the expected number of attempts (considering all attempts to open suitcases—both failed and successful ones).

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