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

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

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