
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, and upper vertex $A$. A line perpendicular to $AF\_2$ passing through $A$ intersects the negative semi-axis of $x$ at point $Q$, and $2 \overrightarrow{F\_1F\_2}+ \overrightarrow{F\_2Q}= \overrightarrow{0}$.
(I) Find the eccentricity of ellipse $C$;
(II) If a circle passing through points $A$, $Q$, and $F\_2$ is tangent to the line $x- \sqrt{3}y-3=0$, find the equation of ellipse $C$;
(III) A line $l$ passing through $F\_2$ intersects the ellipse from part (II) at two distinct points $M$ and $N$. Does the incircle of $\triangle F\_1MN$ have a maximum area? If so, find this maximum value and the equation of line $l$ in this case; otherwise, explain why not.

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