
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 regular triangular pyramid is inscribed in a sphere of radius \( R \), and its height is \( \frac{4R}{3} \). What is the smallest possible area of a cross-section of the pyramid made by a plane passing through a median of the base? Find the ratio of the volumes of the parts into which the plane divides the pyramid in this case.

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