
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 \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0) \) with an eccentricity of \(\frac{1}{2}\), and the distance from the right focus \( F \) to the line \( x-y+2=0 \) is \( 2 \sqrt{2} \). Points \( A_{1} \) and \( A_{2} \) are the left and right vertices of the ellipse \(C\).

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

(2) A line \( l \), passing through \( F \), intersects the ellipse \( C \) at points \( A \) and \( B \) (point \( A \) is above the x-axis). Let \( T \) be the intersection of lines \( A_{1} A \) and \( A_{2} B \). When the y-coordinate of point \( T \) is \( 6 \sqrt{3} \), find the equation of the line \( l \).

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