
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
$S-ABCD$ is a regular quadrilateral pyramid. Points $P$ and $Q$ are the midpoints of $SB$ and $SD$ respectively. The plane through points $A$, $P$, and $Q$ divides the pyramid $SABCD$ into two parts with volumes $V_{1}$ and $V_{2}$ such that $V_{1} < V_{2}$. Find the ratio $\frac{V_{2}}{V_{1}}$.

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