
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\dfrac{\sqrt{3}}{2}$, and a line $l$ with an inclination angle of $30^{\circ}$ passes through the right focus of ellipse $C$ and is tangent to the circle $E$: $x^2 + y^2 = \dfrac{3}{4}$.

$(1)$ Find the equation of ellipse $C$;  
$(2)$ If the line $y = kx + m (k \neq 0)$ is tangent to circle $E$ at point $P$, and intersects ellipse $C$ at points $A$ and $B$, and the ray $OP$ intersects ellipse $C$ at point $Q$, let the area of $\triangle OAB$ and $\triangle QAB$ be $S_1$ and $S_2$ respectively.  
$\quad(1)$ Find the maximum value of $S_1$;  
$\quad(2)$ When $S_1$ reaches its maximum value, find the value of $\dfrac{S_1}{S_2}$.

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