
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 are given a rectangle with length 15 units and width 10 units. The rectangle's perimeter is divided into 50 equally spaced points, with each vertex included and the remaining points dividing each side into equal segments. If $A$, $B$, and $C$ are any three of these points that are not collinear, determine how many different possible positions there are for the centroid of $\triangle ABC$.

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