
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
In triangle $ABC$, a line passing through the centroid $G$ intersects side $AB$ at point $P$ (different from $B$) and side $AC$ at point $Q$ (different from $C$). Let the area of $\triangle APQ$ be $S_1$, and the area of $\triangle ABC$ be $S_2$. Given $\overrightarrow{AP} = p\overrightarrow{PB}$, $\overrightarrow{AQ} = q\overrightarrow{QC}$, find the range of $\frac{S_1}{S_2}$.

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