
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 $F\_2$, respectively, and upper vertex $A$. A line passing through $A$ and perpendicular to $AF\_2$ intersects the negative semi-axis of $x$ at point $Q$, such that $2\overrightarrow{F\_1F\_2} + \overrightarrow{F\_2Q} = \overrightarrow{0}$. A circle passing through points $A$, $Q$, and $F\_2$ is tangent to the line $x - \sqrt{3}y - 3 = 0$.

(I) Find the eccentricity of the ellipse $C$ and its equation.
(II) A line $l$ passing through $F\_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$. Determine whether there exists a maximum area for the incircle of triangle $F\_1MN$. If it exists, find the maximum value and the corresponding line equation; otherwise, explain the reason.

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