
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
In a right pyramid with a square base that has side length $2$ and lateral faces that are equilateral triangles, a cube is perfectly fit inside such that one face lies entirely on the pyramid's square base and each vertex of the opposite cube face touches a lateral face of the pyramid. Find the volume of such a cube.
A) $\frac{2\sqrt{2}}{9}$
B) $\frac{2\sqrt{3}}{9}$
C) $\frac{2\sqrt{6}}{9}$
D) $\sqrt{6}$
E) $\frac{\sqrt{6}}{27}$

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