
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
The lateral surface area of a right circular cone is $p$ square units. The apex angle is $\alpha$. What is the volume of the cone? (Given: $p=160 \ \mathrm{dm}^2$, $\alpha=60^{\circ} 20^{\prime}$.)

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