
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 side of the base of a regular quadrilateral pyramid is $a$, and the lateral edge makes an angle of $30^{\circ}$ with the height. A plane passes through the vertex of the pyramid's base and is perpendicular to the opposite lateral edge. This plane divides the pyramid into two parts. Determine the volume of the part of the pyramid adjacent to the vertex.

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