
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 the equation of circle \\(C\\) is: \\(x^{2}+y^{2}-2x-4y+m=0\\).
\\((1)\\) Find the range of values for \\(m\\);
\\((2)\\) If circle \\(C\\) intersects with the line \\(3x+4y-6=0\\) at points \\(M\\) and \\(N\\), and \\(|MN|=2 \sqrt {3}\\), find the value of \\(m\\);
\\((3)\\) Suppose the line \\(x-y-1=0\\) intersects circle \\(C\\) at points \\(A\\) and \\(B\\), determine whether there exists a real number \\(m\\) such that the circle with diameter \\(AB\\) passes through the origin. If it exists, find the value of \\(m\\); if not, explain why.

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