
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 $n$ at which $a_{n}$ is minimized.

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