
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 $M$: $\frac{x^2}{a^2} + \frac{y^2}{3} = 1 (a > 0)$ has a focus at $F(-1,0)$, and its left and right vertices are $A$ and $B$, respectively. A line $l$ passing through point $F$ intersects the ellipse $M$ at points $C$ and $D$.  
(I) Find the equation of the ellipse;  
(II) When the slope angle of line $l$ is $45^{\circ}$, find the length of segment $CD$;  
(III) Let the areas of $\triangle ABD$ and $\triangle ABC$ be $S_1$ 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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