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

(1) Find the equation of the ellipse $M$;

(2) Denote the areas of triangles $\Delta ABD$ and $\Delta ABC$ 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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