
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 a square has vertices at $(0,0)$, $(15,0)$, $(15,15)$, and $(0,15)$, with 60 points equally distributed around its perimeter. Four points are at the vertices, with the remaining points dividing each side into 15 equal segments. If $P$, $Q$, and $R$ are chosen as any three of these points such that they are not collinear, find the number of different positions possible for the centroid of triangle $\triangle PQR$.

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