
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 triangular pyramid $P-ABC$, where the volume ${V}_{P-ABC}=\frac{4\sqrt{3}}{3}$, $\angle APC=\frac{\pi }{4}$, $\angle BPC=\frac{\pi }{3}$, $PA\perp AC$, $PB\perp BC$, and the plane $PAC$ is perpendicular to the plane $PBC$, find the volume of the circumscribed sphere of the triangular pyramid $P-ABC$.

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