
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
A right cone has a volume of $K = 20 \mathrm{~cm}^{3}$. What is the volume of a three-sided pyramid inscribed in this cone, where the angles of the base triangle are:

$$
\alpha = 25^{\circ} 44^{\prime} 12^{\prime\prime}, \quad \beta = 82^{\circ} 12^{\prime} 40^{\prime\prime}, \quad \gamma = 72^{\circ} 3^{\prime} 8^{\prime\prime}?
$$

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