
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 $M$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with one focus at $F(-1, 0)$, an eccentricity of $e = \frac{1}{2}$, and left and right vertices at $A$ and $B$, respectively. A line $l$ passing through point $F$ intersects the ellipse $M$ at two points $C$ and $D$ (which do not coincide with $A$ and $B$).
(I) Find the equation of the ellipse $M$;
(II) Let the areas of triangles $\triangle ABC$ and $\triangle ABD$ be $S_{1}$ and $S_{2}$, respectively. Find the maximum value of $|S_{1} - S_{2}|$ and the equation of line $l$ in this case.

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