
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 you have a square of side length 20 units with vertices at $(0,0)$, $(20,0)$, $(20,20)$, and $(0,20)$. This square has 80 points equally spaced around its perimeter, with each of the four vertices included and the remaining points dividing each side into 20 equal segments. If $P$, $Q$, and $R$ are any three distinct and non-collinear points among these 80, determine the number of different possible positions for the centroid of $\triangle PQR$.

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