
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
Suppose we have a square with side length 12, and it is divided such that 48 points are equally spaced around the perimeter (with vertices included). Each side is divided into segments that ensure 48 total points. If $P$, $Q$, and $R$ are any three of these points which are not collinear, what is the number of possible positions for the centroid of $\triangle PQR$?

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