
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 circle $A: (x+\sqrt{2})^{2}+y^{2}=12$, and a fixed point $B(\sqrt{2},0)$ inside circle $A$, circle $P$ passes through point $B$ and is tangent to circle $A$ internally.

$(1)$ Find the equation of the trajectory of the center of circle $P$;

$(2)$ If the line $y=kx+2$ intersects the trajectory of point $P$ at points $C$ and $D$, is there a constant $k$ such that the circle with diameter $CD$ passes through the origin $O$? 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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