
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 sides of length $2$ and has lateral faces that are isosceles right triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid, and its opposite face rests such that its vertices touch the midpoints of the lateral faces of the pyramid. Find the volume of the cube.

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