
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 a regular triangular prism \( P-ABC \), there is a hemisphere whose base coincides with the base of the prism, and the three lateral faces of the prism are tangent to the hemisphere. If the radius of the hemisphere is 1, then when the volume of the prism is minimized, the height of the prism is ...

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