
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
An arithmetic progression is a set of the form $\{a, a+d, \ldots, a+k d\}$, where $a, d, k$ are positive integers and $k \geqslant 2$. Thus an arithmetic progression has at least three elements and the successive elements have difference $d$, called the common difference of the arithmetic progression.

Let $n$ be a positive integer. For each partition of the set $\{1,2, \ldots, 3 n\}$ into arithmetic progressions, we consider the sum \( S \) of the respective common differences of these arithmetic progressions. What is the maximal value \( S \) that can attain?

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