
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 fixed point inside a sphere. Three mutually perpendicular lines are drawn from $P$ to intersect the sphere at points $A$, $B$, and $C$, respectively. Point $Q$ is the vertex diagonally opposite to $P$ in the rectangular prism formed by $PA$, $PB$, and $PC$ as the edges. Determine the locus of point $Q$ for all such sets of three lines drawn from point $P$.

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