
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 $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of the semicircle $C$ (with center at point $C$) is $\rho=2\sin \theta$, $\theta\in\left( \frac{\pi}{4}, \frac{3\pi}{4}\right)$.  
(Ⅰ) Find the parametric equation of the semicircle $C$;  
(Ⅱ) The line $l$ intersects the two coordinate axes at points $A$ and $B$, respectively, where $A(0,-2)$. Point $D$ is on the semicircle $C$, and the slope angle of line $CD$ is twice the slope angle of line $l$. If the area of $\triangle ABD$ is $4$, find the Cartesian coordinates of point $D$.

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