
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 $n$, the $n$th square number sum $S_n$ is defined as $S_n = 1^2+2^2+3^2+ \cdots + n^2$. What is the greatest possible value of the greatest common divisor of $3S_n$ and $n+1$?

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