
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \ (a > b > 0)$ with a focal length of $2\sqrt{3}$, and the ellipse $C$ passes through point $A\left(1, \dfrac{\sqrt{3}}{2}\right)$.

$(1)$ Find the equation of ellipse $C$;

$(2)$ If $O$ is the origin, and a line $L$ that does not pass through the origin, $L: y=kx+m$, intersects the ellipse at two distinct points $P(x_1,y_1)$, $Q(x_2,y_2)$, and $y_1y_2=k^2x_1x_2$, find the slope $k$ of line $L$.

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