
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left focus $F$ and upper vertex $A$, a light ray passes through point $A$ and is perpendicular to $AF$. The light ray is reflected by the right directrix of the ellipse, and the reflected light ray is parallel to line $AF$.

1. Determine the eccentricity of the ellipse.
2. If the intersection point of the incident light ray and the right directrix is denoted as $B$, and the circle passing through points $A$, $B$, and $F$ is tangent to the line $3x-y+3=0$, find the equation of the ellipse.

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