
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]


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Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{\sqrt{3}}{2}$, point $A$ is the right vertex of ellipse $C$, point $B$ is a moving point on the ellipse, and $O$ is the origin. If the maximum area of $\triangle OAB$ is $1$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ Let the line $l: y=kx+m$ intersect ellipse $C$ at points $M$ and $N$, with $O$ as the origin. If $k_{OM} \cdot k_{ON} = \frac{5}{4}$, find the maximum area of $\triangle MON$.

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