
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 $\Gamma: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through point $E(\sqrt{3}, \frac{1}{2})$, with an eccentricity of $\frac{\sqrt{3}}{2}$.

(1) Find the equation of ellipse $\Gamma$;

(2) A line $l$ is tangent to circle $O: x^2 + y^2 = b^2$ at point $M$ and intersects ellipse $\Gamma$ at two distinct points $A$ and $B$. Find the maximum area of $\triangle OAB$.

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