
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
Consider the sequence where \( a_n = 105 + n^2 + 3n \) for \( n = 1, 2, 3, \ldots \). For each \( n \), define \( d_n \) as the greatest common divisor of \( a_n \) and \( a_{n+1} \). Determine the maximum value of \( d_n \) as \( n \) varies among the positive integers.

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