
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 $C$ with eccentricity $\frac{\sqrt{3}}{2}$, and the equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$. The ellipse passes through the point $A(2,1)$. <br/>$(1)$ Find the equation of the ellipse $C$. <br/>$(2)$ A line $l$ with slope $k$ that does not pass through point $A$ intersects the ellipse $C$ at points $P$ and $Q$. If the product of the slopes of lines $AP$ and $AQ$ is $\frac{1}{4}$, is $k$ a constant value? If yes, find the value; if not, explain the reason.

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