
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
In the Cartesian coordinate system $xOy$, it is known that the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{3}}{2}$, and its left and right foci are $F_1$ and $F_2$, respectively. A circle centered at $F_1$ with a radius of $3$ intersects with a circle centered at $F_2$ with a radius of $1$, and their intersection points lie on the ellipse $C$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Let the ellipse $E$: $\dfrac{x^2}{4a^2} + \dfrac{y^2}{4b^2} = 1$, and $P$ be any point on the ellipse $C$. A line passing through point $P$ with the equation $y=kx+m$ intersects the ellipse $E$ at points $A$ and $B$, and the ray $PO$ intersects the ellipse $E$ at point $Q$.

$(i)$ Find the value of $\dfrac{OQ}{OP}$;

$(ii)$ Find the maximum value of the area of $\triangle ABQ$.

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