<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 $S-ABCD$ 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 $\frac{V_{2}}{V_{1}}$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
5
$S-ABCD$ 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 $\frac{V_{2}}{V_{1}}$.
test