
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
Inside a regular triangular pyramid, there is a right prism with a rhombus as its base. One face of the prism belongs to the base of the pyramid, and another face to the lateral face of the pyramid. What is the maximum volume of the prism if the edge of the pyramid's base is 2 and the height of the pyramid is \( 2\sqrt{2} \)?

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