
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
Let the left and right foci of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ be $F_1$ and $F_2$, respectively, with the right vertex as $A$ and the top vertex as $B$. It is known that $|AB| = \dfrac{\sqrt{3}}{2}|F_1F_2|$.  
$(1)$ Find the eccentricity of the ellipse;  
$(2)$ Let $P$ be a point on the ellipse other than its vertices, and the circle with diameter $PB$ passes through point $F_1$. A line $l$ passing through the origin $O$ is tangent to this circle. Find the slope of line $l$.

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