
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 that the eccentricity of the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ is $\frac{1}{2}$, and $F\_{1}$, $F\_{2}$ are the left and right foci of the ellipse $C$, respectively. A circle with the origin as its center and the minor axis of ellipse $C$ as its radius is tangent to the line $x - y + \sqrt{6} = 0$.
(I) Find the equation of the ellipse $C$;
(II) If a line $l$ passes through the point $F\_{2}$ and intersects the ellipse $C$ at points $M$ and $N$, find the equation of the line $l$ when the area of the triangle $F\_{2}MN$ is maximized.

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