
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
Given a sphere with radius $R$, three mutually perpendicular segments $PA, PB, PC$ are drawn terminating on the sphere and having one common vertex at $P$. Consider the rectangular parallelepiped of which $PA, PB, PC$ are coterminal edges. Find the locus of the point $Q$ that is diagonally opposite $P$ in the parallelepiped when $P$ and the sphere are fixed.

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