
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 solid has a square base where the diagonal of the square base equals \(4\sqrt{2}\). Each vertex of the square base is extended vertically upward to a point such that these vertices form a smaller square parallel to the base. The side length of the top square is \(\sqrt{2}\), positioned such that the diagonals of the base and the top square align perpendicularly. If the height from the base to the top square is 2, determine the volume of the solid.

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