
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) If point A $(m, -2)$ is inside circle C, find the range of values for $m$;  
(2) If $m=4$, (i) let P $(x, y)$ be a moving point on circle C, find the maximum and minimum values of $(x-4)^2+(y-2)^2$; (ii) determine whether there exists a line $l$ with a slope of 1 such that the chord AB cut by circle C, with AB as the diameter, passes through the origin. If it exists, write the equation of line $l$; if not, explain why.

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