
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 curve C is $x^2+y^2-3x=0$ ($\frac{5}{3}<x\leq3$).

(1) Find the coordinates of the center of the circle where curve C lies.  
(2) Determine whether there exists a real number $k$ such that the line $L: y=k(x-4)$ intersects curve C at only one point. If it exists, find the range of $k$; if not, explain why.

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