
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 $C$ with an eccentricity of $\frac{\sqrt{3}}{2}$, and a line passing through the top vertex and the left focus has an inclination angle of $\frac{\pi}{6}$. A line $l$ passes through point $E(-1,0)$ and intersects ellipse $C$ at points $A$ and $B$.  
(1) Find the standard equation of ellipse $C$;  
(2) Does the area of $\triangle AOB$ have a maximum value? If yes, find this maximum value; if no, explain why.

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