
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 \( P \) be a point inside a regular tetrahedron \( T \) of unit volume. The four planes passing through \( P \) and parallel to the faces of \( T \) partition \( T \) into 14 pieces. Let \( f(P) \) be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for \( f(P) \) as \( P \) varies over \( T \).

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