
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, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \rho = -4\sqrt{3}\sin \theta$ intersects with circle $C$ at points $A$ and $B$.  
(i) Find the polar equation of line $AB$.  
(ii) If a line $C_2$ passing through point $C(2,0)$ with the parametric equations $\begin{cases} x=2+ \frac{\sqrt{3}}{2}t, \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter) intersects line $AB$ at point $D$ and intersects the $y$-axis at point $E$, calculate the ratio $|CD| : |CE|$.

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