
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \, (a > b > 0)$ has an eccentricity of $\frac{\sqrt{3}}{2}$. The left and right foci of the ellipse are $F_1$ and $F_2$ respectively. A circle centered at $F_1$ with a radius of $3$ intersects with another circle centered at $F_2$ with a radius of $1$, and their intersection points lie on the ellipse $C$.  
(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) Suppose ellipse $E: \frac{x^2}{4a^2} + \frac{y^2}{4b^2} = 1$, and $P$ is any point on ellipse $C$. A line passing through point $P$ with the equation $y=kx+m$ intersects ellipse $E$ at points $A$ and $B$, and a ray from $P$ toward the origin $O$ intersects with ellipse $E$ at point $Q$.  
(i) Find the value of $| \frac{OQ}{OP} |$;  
(ii) Find the maximum value of the area of $\triangle ABQ$.

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