
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, passing through the point $(\sqrt{3}, \dfrac{1}{2})$, with foci $F_1$ and $F_2$, and the eccentricity of the ellipse is $\dfrac{\sqrt{3}}{2}$, $O$ is the origin of coordinates.

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

$(2)$ A line $l$ passing through the left focus $F_1$ intersects the ellipse at points $P$ and $Q$ (different from the left and right vertices), find the maximum value of the radius of the inscribed circle of $\triangle PQF_2$.

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