
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
Given a pyramid with an equilateral triangular base (each side has length 3) and isosceles triangular lateral faces, the apex of the pyramid is directly above the centroid of the base at a height of 9 units from the base. Find the edge length of the cube positioned within the pyramid such that its one vertex is at the centroid of the base and one of its body diagonals is aligned from this point towards the apex of the pyramid.

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