
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
There are \( n \) identical-looking keys, only one of which can open the lock on the main door. Assume that drawing the keys is mutually independent and equiprobable, and each key is not returned after being tried. Find the distribution and the expected value \( E(\varepsilon) \) of the number of trials \( \varepsilon \).

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