
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 $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a \gt b \gt 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, and focal distance $2\sqrt{3}$, where point $Q(\sqrt{3}$,$-\frac{1}{2})$ lies on ellipse $C$. <br/>$(1)$ Let $P$ be a moving point on $C$, find the range of $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}$; <br/>$(2)$ A line $l$ passing through the right focus $F_{2}$ of $C$ with a non-zero slope intersects $C$ at points $M$ and $N$, find the maximum value of the area of the incircle of $\triangle F_{1}MN$.

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