
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
There are $n$ marbles, and they are divided into two piles in any manner. Determine the product of the numbers of marbles in the two piles. Then, take one of the piles and divide it into two piles in any manner. Determine the product of the numbers of marbles in these two new piles. Repeat this process: each time take any pile and divide it into two piles, determine the product of the numbers of marbles in the two piles, and continue until no pile can be further divided. Denote the sum of all these products as $S$. Determine the possible values of $S$.

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