
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
For all positive integers \( n \), define the \( n \)th square number \( S_n \) as \( S_n = n^2 \). Determine the greatest possible value of the greatest common divisor of \( 6S_n \) and \( n+1 \).

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