
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 64 points equally spaced around the perimeter of a square, such that there are four points at the vertices and the remaining points divide each side into 15 equal segments. If $P$, $Q$, and $R$ are selected as any three points among these that are not collinear, how many different positions are possible for the centroid of $\triangle PQR$?

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