
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 points $A(6,0)$, $B(1,5)$, and the center of the circle is on the line $l$: $2x-7y+8=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ A line passing through point $M(1,2)$ intersects circle $C$ at points $A$ and $B$. Is there a fixed point $N$ on the line $y=2$ such that $K_{AN}+K_{BN}=0$ always holds? If it exists, find the coordinates of point $N$; if not, explain why.

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