
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 $M: \frac{x^2}{a^2} + \frac{y^2}{3} = 1$ with $(a > 0)$, one of its focal points is $F(-1,0)$, and its left and right vertices are $A$ 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;

(2) When the inclination of line $l$ is $45^{\circ}$, find the length of the line segment $CD$;

(3) Let the areas of triangles $ABD$ and $ABC$ be denoted by $S_1$ and $S_2$ respectively, find the maximum value of $|S_1 - S_2|$.

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