
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
Let $P(e, \pi, \sqrt{5})$ be a point inside a cube with 8 vertices $(x, y, z)$, where $x, y, z$ are either 0 or 6. Find the volume of the intersection of the cube and the polyhedron formed by the 6 points symmetric to $P$ with respect to each face of the cube.

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