
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
Consider an ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, right vertex $A$, and upper vertex $B$. It is known that $|AB| = \frac{\sqrt{3}}{2} |F\_1F\_2|$.

(1) Find the eccentricity of the ellipse.

(2) Let $P$ be a point on the ellipse other than its vertices. A circle passing through point $F\_1$ has $PB$ as its diameter. 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}
user