
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 $C$ with its center at the origin and its foci on the x-axis, the length of the major axis is $\sqrt{3}$ times the length of the minor axis. The shortest distance from a point on the ellipse to a focus is $\sqrt{3} - \sqrt{2}$.
(1) Find the equation of ellipse $C$;
(2) If the line $l: y = kx + b$ is tangent to the circle $O: x^2 + y^2 = \frac{3}{4}$ and intersects ellipse $C$ at points $A$ and $B$, find the equation of line $l$ when the area of $\triangle AOB$ is maximized.

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