
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 triangular number \(T_n\) is defined as \(T_n = 1+2+3+ \cdots + n\). What is the greatest possible value of the greatest common divisor of \(6T_n\) and \(n-2\)?

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