
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 $F(-1, 0)$. Points $A$ and $B$ are the left and right vertices of the ellipse's major axis, respectively. A line $l$ passes through $F$ and intersects the ellipse at distinct points $C$ and $D$.

1. Find the equation of the ellipse $M$;
2. When the line $l$ has an angle of $45^{\circ}$, find the length of the line segment $CD$;
3. Let $S_1$ and $S_2$ represent the areas of triangles $\Delta ABC$ and $\Delta ABD$, respectively. Find the maximum value of $|S_1 - S_2|$.

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