
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 unit cube is cut once along a diagonal of one face and then along the diagonal of the opposite parallel face creating four triangular prisms. One of these prisms is cut again along a plane passing through one of its longest edges and the midpoint of its opposite edge. This results in six pieces in total. What is the volume of the triangular prism that includes vertex $W$ on the original cube?

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