
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 prism \( P-ABC \) where the three face angles at the top vertex are all \( 60^\circ \), i.e., \( \angle APB = \angle BPC = \angle CPA = 60^\circ \), and the areas of the three lateral faces are \( \frac{\sqrt{3}}{2}, 2, \text{and}\, 1 \), find the volume of the triangular prism.

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