
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
Given the parabola $C_{1}$: $x^{2}=4y$ with focus $F$ also being a focus of the ellipse $C_{2}$: $\frac{y^2}{a^2}+\frac{x^2}{b^2}=1\left(a \gt b \gt 0\right)$, and the common chord length of $C_{1}$ and $C_{2}$ is $2\sqrt{6}$.  
$(1)$ Find the equation of the ellipse $C_{2}$;  
$(2)$ A line $l$ with slope $k$ passes through points $A$ and $B$ where it intersects the parabola $C_{1}$, and points $C$ and $D$ where it intersects the ellipse $C_{2}$, and $\overrightarrow{AC}$ and $\overrightarrow{BD}$ are in the same direction.  
$(i)$ Determine the shape of $\triangle OAB$ as the line $l$ rotates around point $F$;  
$(ii)$ If $|AC|=|BD|$, find the slope of the line $l$.

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