
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the left focus $F(-1,0)$, left, right, and top vertices denoted as $A$, $B$, $C$ respectively, and $\overrightarrow{CF} \cdot \overrightarrow{CB} = 1$.  
$(1)$ Find the equation of the ellipse $E$;  
$(2)$ If the line $l: y = kx - 2$ intersects the ellipse $E$ at points $M$ and $N$, find the maximum area of $\triangle OMN$ and the equation of the line $l$ that achieves this maximum area.

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