
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 are given 52 points equally spaced around the perimeter of a rectangle, with four points at the vertices and the remaining points dividing each side as follows: two longer sides with each 16 points, and two shorter sides with each 8 points. If $P$, $Q$, and $R$ are any three of these points which are not collinear on the rectangle, determine 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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