
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 $F$, which is also a focus of the ellipse $C_{2}: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1$ $(a > b > 0)$. The common chord of $C_{1}$ and $C_{2}$ has a length of $2\sqrt{6}$.
(Ⅰ) Find the equation of $C_{2}$;
(Ⅱ) A line $l$ passes through point $F$ and intersects $C_{1}$ at points $A$ and $B$, and intersects $C_{2}$ at points $C$ and $D$. It's given that the vectors $\overrightarrow{AC}$ and $\overrightarrow{BD}$ have the same direction. If $|AC|=|BD|$, find the slope of line $l$.

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