
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 right triangular prism ABC-A$_{
m 1}$B$_{
m 1}$C$_{
m 1}$ with a base edge length of $2\sqrt {3}$ and a height of 3, circle O is the inscribed circle of triangle ABC, and point P is any point on circle O. The volume of the circumscribed sphere of the triangular pyramid P-A$_{
m 1}$B$_{
m 1}$C$_{
m 1}$ is $\_\_\_\_\_\_.$

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