
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 line \\(l\\) passing through the origin intersects with circle \\(C_{1}\\): \\(x^{2}+y^{2}-6x+5=0\\) at two distinct points \\(A\\) and \\(B\\).

\\((1)\\) Find the coordinates of the center of circle \\(C_{1}\\);  
\\((2)\\) Find the equation of the trajectory \\(C\\) of the midpoint \\(M\\) of segment \\(AB\\);  
\\((3)\\) Determine whether there exists a real number \\(k\\) such that the line \\(L\\): \\(y=k(x-4)\\) intersects with curve \\(C\\) at only one point. If it exists, find the range of \\(k\\); if not, explain why.

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