
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 pyramid with an equilateral triangle base, where each side is 2 units long, contains a cube. The height from the base to the apex of the pyramid is 3 units. The cube is oriented such that one vertex touches the centroid of the base triangle, and the face opposite to this vertex touches the lateral faces of the pyramid. Find the maximum volume of this cube.

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