
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
Consider a square in the coordinate plane with vertices at $(0,0)$, $(20,0)$, $(20,20)$, and $(0,20)$. Along the perimeter of this square, 80 points are equally spaced so that each side is divided into 20 segments of equal length. Three points $P$, $Q$, and $R$, which are not collinear, are selected from these 80 points. Determine the number of possible positions for the centroid of $\triangle PQR$.

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