
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 pyramid with a regular hexagonal base, each side having a length of 2 units. The pyramid has six lateral faces that are isosceles triangles, with each triangle having a height of 3 units. A cube is perfectly inscribed in the pyramid so that each vertex of the cube is either on the base or touches a point on the lateral faces of the pyramid. What is the volume of this cube?
A) $2\sqrt{2}$
B) $4$
C) $\sqrt{2}$
D) $\frac{1}{\sqrt{2}}$
E) $\frac{\sqrt{2}}{2}$

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