
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 the ellipse $\Gamma$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passes through the point $E(\sqrt{3}, \dfrac{1}{2})$, and its eccentricity is $\dfrac{\sqrt{3}}{2}$.

$(1)$ Find the equation of the ellipse $\Gamma$;  
$(2)$ A line $l$ is tangent to the circle $O$: $x^{2} + y^{2} = b^{2}$ at point $M$, and intersects the ellipse $\Gamma$ at two distinct points $A$ and $B$. Find the maximum value of $|AB|$.

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