
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 an ellipse $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a＞b＞0)$ passing through the point $({\sqrt{3},\frac{1}{2}})$, with its right focus at $F({\sqrt{3},0})$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ Let the right vertex of the ellipse $C$ be $A$. If points $P$ and $Q$ lie on the ellipse $C$ and the product of the slopes of lines $AP$ and $AQ$ is $\frac{1}{{20}}$, find the maximum area of $\triangle APQ$.

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