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

1. Find the equation of the ellipse (Γ).
2. Denote the areas of triangles $ABC$ and $ABD$ as $S_1$ and $S_2$, respectively. Find the range of values for $S_1 - S_2$.

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