
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 $C$ passes through two points $M(-3,3)$ and $N(1,-5)$, and the center of the circle $C$ is on the line $2x-y-2=0$.

(Ⅰ) Find the standard equation of circle $C$;  
(Ⅱ) Line $l$ passes through point $(-2,5)$ and intersects circle $C$ at two distinct points $A$ and $B$. If the slope $k$ of line $l$ is greater than $0$, find the range of values for $k$;  
(Ⅲ) Under the condition of (Ⅱ), determine whether there exists a line $l$ such that the perpendicular bisector of chord $AB$ passes through point $P(3,-1)$. If it exists, find the equation of line $l$; if not, explain why.

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