
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 the equation $\frac{x^2}{a^2} + \frac{y^2}{3} = 1$ where $a > \sqrt{3}$, let $F$ be the right focus, and $A$ be the right vertex of the ellipse. It is known that $\frac{1}{|OF|} + \frac{1}{|OA|} = \frac{3e}{|FA|}$, with $O$ being the origin and $e$ the eccentricity of the ellipse.
$(1)$ Find the equation of the ellipse.
$(2)$ Suppose a line $l$ passing through point $A$ intersects the ellipse at point $B$ (which is not on the x-axis), a line perpendicular to $l$ goes through point $M$ (the intersection with $l$) and point $H$ (the intersection with the y-axis). If $BF \perp HF$ and $\angle MOA = \angle MAO$, find the slope of line $l$.

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