
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 rectangle with vertices at $(0,0)$, $(12,0)$, $(12,8)$, and $(0,8)$, with 48 points equally spaced along its perimeter. If $A$, $B$, and $C$ are any three non-collinear points among these, determine how many different possible positions there are for the centroid of $\triangle ABC$.

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