
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
In the right triangular pyramid $P-ABC$, the side edges $PA$, $PB$, $PC$ are pairwise perpendicular. $Q$ is a point inside the base triangle $\triangle ABC$. If the distance from point $Q$ to the three side faces is $3\sqrt{2}$, $4\sqrt{2}$, and $5\sqrt{2}$ respectively, find the volume of the sphere with diameter $PQ$.

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