
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
In the Cartesian coordinate system, there is a circle \\(C\\) with its center on the line \\(x-2y=0\\) and passing through point \\(A(4,0)\\), but not through the origin. The chord formed by the intersection of the line \\(4x-3y=0\\) and circle \\(C\\) has a length of \\(4\\).
\\((1)\\) Find the general equation of circle \\(C\\);  
\\((2)\\) If a ray emitted from point \\(M(-4,1)\\) reflects off the \\(x\\)-axis and the reflected ray passes exactly through the center of circle \\(C\\), find the equation of the line of the reflected ray (expressed in general form).

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