
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 a regular hexagonal base where each side has a length of 2 units, the height is adjusted so that the lateral faces are equilateral triangles, and a cube and a regular tetrahedron are placed within the pyramid such that one of the cube's faces and one face of the tetrahedron lie entirely on the pyramid's base. Find the volume of the cube if one edge of the cube touches one of the lateral faces of the pyramid and one edge of the tetrahedron also touches a different lateral face of the pyramid.

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