
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 $A,B$ are the endpoints of the major axis of the ellipse $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$, $F(\sqrt{3},0)$ is one of its foci, and $\overrightarrow{AF}\cdot \overrightarrow{BF}=-1$.

(1) Find the equation of the ellipse $E$;

(2) If line $l$ intersects the ellipse $E$ at points $M,N$, and a circle passing through the origin $O$ has $MN$ as its diameter, find the maximum value of the area of $\Delta OMN$.

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