
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 with eccentricity $e = \frac{\sqrt{3}}{2}$ and the fact that joining the four vertices of the ellipse forms a rhombus with area 4.
(1) Find the equation of the ellipse;
(2) Suppose a line $l$ intersects the ellipse at two distinct points A and B. Point A has coordinates $(-a,0)$. If $|AB| = \frac{4\sqrt{2}}{5}$, find the value of the slope $k$ of the line $l$.

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