
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
Given is a regular pentagonal prism and a square plate \(ABCD\), all of whose edges are equal and of unit length. The prism is placed twice successively on the (horizontally placed) plate by its side face, first aligning the \(AB\) edge and then aligning the \(AD\) edge with a side edge of the prism.

What is the combined volume of the space occupied by the two positions of the prism?

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