
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 ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{3}=1(a > 0)$ has one of its foci at $(-1,0)$, with its left and right vertices at $A$ and $B$, respectively. A line $l$ passing through point $F$ intersects the ellipse $M$ at points $C$ and $D$.

(1) Find the equation of the ellipse;

(2) Denote the areas of triangles $\triangle ABD$ and $\triangle ABC$ as $S_{1}$ and $S_{2}$, respectively. Find the maximum value of $|S_{1}-S_{2}|$.

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