
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 an equilateral triangle with vertices at \( (0, 0) \), \( (6, 0) \), and \( (3, 3\sqrt{3}) \), and we place 36 points equally spaced along the perimeter (including the vertices and dividing each side into 12 congruent sections). If \( P \), \( Q \), and \( R \) are any three non-collinear points among these, how many different possible positions are there for the centroid of \( \triangle PQR \)?

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