
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 that $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and point $P\left(\frac{2 \sqrt{6}}{3}, 1\right)$ lies on the ellipse $C$. The orthocenter of $\triangle F_{1} P F_{2}$ 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 $C$. A line $l$ that passes through $F_{2}$ intersects the ellipse $C$ at points $D$ and $E$. Denote the slopes of lines $A D$ and $A E$ as $k_{1}$ and $k_{2}$ respectively. If $k_{1}+k_{2}=-\frac{1}{2}$, find the equation of the line $l$.

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