
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\\) whose center is 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)\\) Is there a positive real number \\(r\\) such that there is exactly one circle within the moving circle \\(C\\) that is externally tangent to the circle \\(O:x^{2}+y^{2}=r^{2}\\)? If it exists, find the value of \\(r\\); if not, explain why.

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