
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
Given $n$ locked suitcases and $n$ keys, where each suitcase is unlocked by exactly one key and each key unlocks exactly one suitcase. Our goal is to determine which key opens which suitcase. An attempt consists of trying to unlock a suitcase with a specific key. Determine the smallest number $p(n)$ for which there exists a procedure that ensures we know all $n$ suitcase-key pairs after at most $p(n)$ attempts.

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