
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. The distance from the right focus to the line $l\_1: 3x + 4y = 0$ is $\frac{3}{5}$.
(I) Find the equation of ellipse (C);
(II) Suppose line $l\_2: y = kx + m (km \neq 0)$ intersects the ellipse (C) at points A and B, and the midpoint of segment AB is exactly on line $l\_1$. Find the maximum value of the area S of triangle OAB (where O is the origin of the coordinate plane).

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