
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 triangle in the coordinate plane with vertices at $(0,0)$, $(38,0)$, and $(18,30)$. A triangular pyramid is formed by joining the midpoints of the triangle's sides with an external apex directly above the centroid of the triangle. Calculate the volume of this pyramid.

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