
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
The sequence \(101, 104, 116, \cdots\) is given by the general term \(a_{n} = 100 + n^{2}\), where \(n = 1, 2, 3, \cdots\). For each \(n\), let \(d_{n}\) denote the greatest common divisor of \(a_{n}\) and \(a_{n+1}\). Find the maximum value of \(d_{n}\) for all positive integers \(n\).

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