
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 that the focal length of the ellipse $C:\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$ is $2\sqrt{6}$, and the sum of the distances from any point on the ellipse $C$ to the two foci of the ellipse is $6$.
(I) Find the equation of the ellipse $C$;
(II) Let line $l:y=kx-2$ intersect with ellipse $C$ at two points $A$ and $B$, point $P(0,1)$, and $|PA|=|PB|$, find the equation of line $l$.

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