
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 with center $C$ that passes through points $A(2,4)$ and $B(3,5)$, and the center $C$ is on the line $2x-y-2=0$.

(Ⅰ) Find the equation of the circle with center $C$;

(Ⅱ) If the line $y=kx+3$ always has common points with the circle, find the range of the real number $k$.

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