
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 that the volume of a regular triangular pyramid $P-ABC$ is $\frac{1}{12}$, the center of its circumscribed sphere is $O$, and it satisfies $\vec{OA} + \vec{OB} + \vec{OC} = \vec{0}$, find the radius of the circumscribed sphere of the regular triangular pyramid $P-ABC$.

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