
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 circle \\(C\\) that passes through \\(A(1,-1)\\), \\(B(5,3)\\), and is bisected in area by the line \\(m\\): \\(3x-y=0\\).  
\\((\\)I\\()\\) Find the equation of circle \\(C\\);  
\\((\\)II\\()\\) If a line \\(l\\) passing through point \\(D(0,-1)\\) with slope \\(k\\) has two distinct common points with circle \\(C\\), find the range of the real number \\(k\\).

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