
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  $n\geq 2$  be an integer. For each  $k$ -tuple of positive integers  $a_1, a_2, \ldots, a_k$  such that  $a_1+a_2+\cdots +a_k=n$ , consider the sums  $S_i=1+2+\ldots +a_i$  for  $1\leq i\leq k$ . Determine, in terms of  $n$ , the maximum possible value of the product  $S_1S_2\cdots S_k$ .

*Proposed by Misael Pelayo*

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