
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
A tetrahedral pyramid has an equilateral triangular base with side length 3 and has lateral faces that are equilateral triangles. A cube is placed within the tetrahedral pyramid such that one vertex of the cube touches the centroid of the triangular base and its opposite edges touch the midpoints of the slant edges of the pyramid. Calculate the volume of this cube.

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