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

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