
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 $T$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{3}}{2}$. If ellipse $T$ intersects with circle $P: (x + \dfrac{3}{2})^{2} + y^{2} = 1$ at points $M$ and $N$, and the arc of circle $P$ inside ellipse $T$ has a length of $\dfrac{2}{3}\pi$.

$(1)$ Find the values of $a$ and $b$;  
$(2)$ Draw two lines $AC$ and $BD$ through the center of ellipse $T$, intersecting ellipse $T$ at points $A$, $C$, and $B$, $D$ respectively. Let the slope of line $AC$ be $k_{1}$ and the slope of line $BD$ be $k_{2}$, and $k_{1}k_{2} = \dfrac{1}{4}$.  
$\quad(①)$ Find the slope of line $AB$;  
$\quad(②)$ Find the range of the area of quadrilateral $ABCD$.

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