
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 receives 10 suitcases and 10 keys, where each key opens only one suitcase. The task is to match each key to the correct suitcase. A worker laments that he may need to try many keys for each suitcase before finding the right one. 

An employee claims the maximum number of attempts is $10+9+8+\ldots+2+1=55$. Another employee suggests that the maximum number of attempts is $9+8+\ldots+1=45$, assuming if a key does not fit in nine suitcases, it will fit in the tenth. 

The average number of attempts is expected to be half of the maximum, or 22.5. However, Igor Fedorovich Akulich from Minsk questions the reasoning behind this and suggests it may not be accurate.

The goal is to find the expected number of attempts required to open all suitcases, accounting for both successful and unsuccessful attempts.

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