
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 \( F_{1} \) and \( F_{2} \) as the left and right foci of the ellipse \( C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with \( a > b > 0 \), and the point \( P\left(\frac{2 \sqrt{6}}{3}, 1\right) \) lies on ellipse \( C \). The orthocenter of triangle \( \triangle F_{1} P F_{2} \) is \( H\left(\frac{2 \sqrt{6}}{3}, -\frac{5}{3}\right) \).

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

(2) Let \( A \) be the left vertex of the ellipse \( C \). The line \( l \) passes through \( F_{2} \) and intersects the ellipse \( C \) at points \( D \) and \( E \). Let the slopes of lines \( AD \) and \( AE \) be \( k_{1} \) and \( k_{2} \), respectively. Given \( k_{1} + k_{2} = -\frac{1}{2} \), find the equation of the line \( l \).

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