
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 $xoy$, it is known that the eccentricity of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a > b > 0)$ is $\frac{\sqrt{3}}{2}$, and the left and right foci are ${F}_{1},{F}_{2}$ respectively. A circle with ${F}_{1}$ as the center and a radius of $3$ intersects with a circle with ${F}_{2}$ as the center and a radius of $1$. The intersection points are on the ellipse $C$.

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

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

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

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

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