
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 the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1} = p\), \(a_{2} = p + 1\), and \(a_{n+2} - 2a_{n+1} + a_{n} = n - 20\), where \(p\) is a given real number and \(n\) is a positive integer, find the value of \(n\) such that \(a_{n}\) is minimized.

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