
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 60 points equally spaced around the perimeter of a rectangle with length 15 units and width 10 units. Four of these points are at the vertices, and the remaining points divide each side into parts with 14 points on the longer sides and 8 points on the shorter sides. If $P$, $Q$, and $R$ are any three of these points that are not collinear, 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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