
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 $\Gamma$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), it passes through point $P(1, -\frac{\sqrt{3}}{2})$, and its eccentricity is $\frac{\sqrt{3}}{2}$. The left focus is $F$, and the left and right vertices are $A$ and $B$, respectively. The line $l$ passing through point $F$ intersects with ellipse $\Gamma$ at points $C$ and $D$.

(1) Find the equation of the ellipse $\Gamma$;

(2) Let $S_1$ and $S_2$ represent the areas of $\Delta ABC$ and $\Delta ABD$, respectively. Find the maximum value of $|S_1 - S_2|$.

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