
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 rectangular coordinate system $(xOy)$, an ellipse $(C)$ is given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$. The left and right foci of the ellipse are $F_1$ and $F_2$, respectively. A circle with center $F_1$ and radius $3$ intersects a circle with center $F_2$ and radius $1$ at points that lie on the ellipse $(C)$.

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

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

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

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

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