
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 $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a \gt b \gt 0\right)$, where the upper vertex $E$ and the left and right foci $F_{1}$, $F_{2}$ of the ellipse form a right-angled triangle with an area of $1$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ A line $l$ passing through point $F_{2}$ intersects the ellipse $C$ at points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$. If $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}=3$, find the equation of the line $l$.

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