
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 place 36 points equally spaced around the perimeter of a square, such that each vertex of the square is one of these points and the remaining points divide each side into eight equal segments.  If $A$, $B$, and $C$ are any three of these points which are not collinear, determine how many distinct possible positions there are for the centroid of $\triangle ABC$.

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