
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 regular tetrahedron $T$ with unit volume is scaled by a factor of 2 from a point $P$ on one of its axes of symmetry. The volume of the shared region between the original and the scaled tetrahedra is \( \frac{1}{8} \). How far is the center of $T$ from $P$?

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