
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
Given a regular quadrilateral pyramid $S-ABCD$, points $P$ and $Q$ are the midpoints of $SB$ and $SD$ respectively. Find the ratio of the volume of the larger part to the smaller part of the pyramid after the plane passing through points $A$, $P$, and $Q$ divides the pyramid.

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