
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 a moving circle $C$ with a radius of $5$ and its center on the line $l$: $x-y+10=0$. (1) If the moving circle $C$ passes through the point $(-5,0)$, find the equation of circle $C$; (2) Does there exist a positive real number $r$ such that there is a unique circle in the moving circle $C$ that is externally tangent to the circle $O$: $x^{2}+y^{2}=r^{2}$? If such an $r$ exists, find it; otherwise, explain why not.

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