
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 $xOy$, a moving point $P$ has a distance to $F(2\sqrt{2}, 0)$ and its distance to the line $L: x=3\sqrt{2}$ in the ratio of $\frac{\sqrt{6}}{3}$.  
(1) Find the equation of the trajectory of point $P$.  
(2) Determine whether there exists a line $l: y=kx-2$ ($k\neq 0$) such that line $l$ intersects the trajectory of point $P$ at two distinct points $M$ and $N$, and $|\overrightarrow{AM}|=|\overrightarrow{AN}|$, where $A(0, 2)$. If it exists, find the value of $k$; if not, explain why.

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