
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 two foci ${F}_{1},{F}_{2}$, and an eccentricity of $\frac{\sqrt{3}}{2}$. $O$ represents the coordinate origin.

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

(2) A line $l$ passing through the left focus ${F}_{1}$ intersects the ellipse at points $P$ and $Q$ (which are not the left and right vertices), find the maximum value of the inradius of $\triangle PQ{F}_{2}$.

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