
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 line $l$ passing through the origin and a circle $C$: $x^2+y^2-8x+12=0$ intersecting at two distinct points $A$, $B$.
   (1) Find the trajectory equation $M$ of the midpoint $P$ of the line segment $AB$.
   (2) Determine if there exists a real number $k$ such that the line $l_1$: $y=k(x-5)$ intersects the curve $M$ at exactly one point. If it exists, find the range of $k$; otherwise, explain the reason.

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