
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{\sqrt{3}}{2}$, where the left and right foci are $F_{1}$ and $F_{2}$ respectively, the top vertex is $P$, and $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}} = -2$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ The line $l: y=kx+m(m>0)$ intersects the ellipse $C$ at points $A$ and $B$, with the origin $O$. Find the value of $k$ for which $|OA|^2+|OB|^2$ remains constant, and determine the maximum area of triangle $\triangle AOB$ at this value of $k$.

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