
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
An ellipse is 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 $F_2$, respectively, and right and upper vertices $A$ and $B$. 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. The circle with diameter $PB$ passes through point $F_1$, and the line $l$ through point $F_2$ is tangent to this circle at point $M, |MF_2| = 2\sqrt{2}$. Find the equation of the ellipse.

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