
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 hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) with \( a>0 \) and \( b>0 \), its eccentricity \( e=\sqrt{3} \) and the distance from its left focus \( F_{1} \) to the asymptote is \( \sqrt{2} \).

(1) Find the equation of the hyperbola \( C \).

(2) If the line \( l \) passing through the point \( D(2,0) \) intersects the hyperbola \( C \) at points \( A \) and \( B \), and the circle with diameter \( AB \) passes through the origin \( O \), find the equation of the line passing through \( A \) and \( B \).

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