
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 has a square base with side length 2 and lateral faces that are equilateral triangles. A cube is placed within the pyramid such that one face of the cube is on the base of the pyramid and its opposite face has all its vertices touching the lateral faces of the pyramid. Find the volume of the cube.
**A)** $\frac{3}{8}$  
**B)** $\frac{3\sqrt{2}}{8}$  
**C)** $\frac{3\sqrt{3}}{8}$  
**D)** $\frac{\sqrt{3}}{9}$  
**E)** $\frac{\sqrt{2}}{9}$

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