
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) with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) passing through point $P(1, \frac{3}{2})$ and an eccentricity of $\frac{1}{2}$.

(I) Find the standard equation of ellipse (C);

(II) Let ($F\_1$, $F\_2$) be the left and right foci of ellipse (C), respectively. Line $l$ passes through $F\_2$ and intersects ellipse (C) at two distinct points $M$ and $N$. Let $S$ be the area of the incircle of triangle $F\_1MN$. Find the equation of line $l$ when $S$ is at its maximum value, and determine this maximum value.

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