
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 with vertices at $(0,0)$, $(12,0)$, $(12,12)$, and $(0,12)$. This square's perimeter is divided into 48 equal intervals, adding more points than the original setting; four are vertices, and the remaining points split each side into twelve equal segments. How many vertices exist for the centroid of a triangle formed by choosing any three non-collinear points among these?

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