
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 C is defined 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 an eccentricity of $$\frac{\sqrt{3}}{2}$$. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse C at a segment of length 1.

1. Find the equation of ellipse C.
2. Let line $l: y = kx + m$ be a tangent line to a circle centered at the origin O with radius $$\frac{2\sqrt{5}}{5}$$, intersecting ellipse C at two distinct points A and B. Find the maximum area of triangle AOB.

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