
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 points \\(A(0,2)\\) and \\(B(2,-2)\\), and the center of circle \\(C\\) is on the line \\(l\\): \\(x-y+1=0\\).
\\((1)\\) Find the equation of circle \\(C\\);
\\((2)\\) If line \\(m\\) passes through point \\((1,4)\\) and the chord intercepted by circle \\(C\\) has a length of \\(6\\), find the equation of line \\(m\\).

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