
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 volume of a regular quadrangular pyramid is \( V \), and the angle between a lateral edge and the base plane is \( 30^{\circ} \). Consider regular triangular prisms inscribed in the pyramid such that one of the lateral edges lies along the diagonal of the pyramid's base, one of the lateral faces is parallel to the base of the pyramid, and the vertices of this face are on the lateral edges of the pyramid. Find: a) the volume of the prism whose lateral face plane divides the height of the pyramid in the ratio 2:3, starting from the vertex; b) the maximum possible volume of the considered prisms.

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