
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 the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where $F_1$ and $F_2$ are the left and right foci of the ellipse, and $A$ is the upper vertex of the ellipse. The line $AF_2$ intersects the ellipse at another point $B$.

(1) If $\angle F_1AB = 90^{\circ}$, find the eccentricity of the ellipse.

(2) If $\overrightarrow{F_2} = 2\overrightarrow{F_1}$ and $\overrightarrow{F_1} \cdot \overrightarrow{AB} = \frac{3}{2}$, find the equation of the ellipse.

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