
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 two circles ${C_1}:{({x-2})^2}+{y^2}=18$, ${C_2}:{({x+2})^2}+{y^2}=2$, a moving circle $M$ is inside circle $C_{1}$ and is tangent to circle $C_{1}$ internally and tangent to circle $C_{2}$ externally.  
$(1)$ Find the equation $C$ of the locus of the center of circle $M$.  
$(2)$ Determine whether there exists a circle with its center at the origin, such that any tangent line of this circle always intersects the locus equation $C$ at two points $M$ and $N$, and satisfies $OM\bot ON$. If it exists, find the equation of this circle; if not, explain the reason.

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