
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]


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A unit cube is cut three times diagonally from each vertex on one face to the opposite vertex of the opposite face, creating four triangular prisms. The cube is then cut along three more planes parallel to the sides of the cube and passing through the center, creating a total of 27 smaller pieces. What is the volume of the piece that contains vertex \( W \)?

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