
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 that the circle $C$ passes through the points $E(0,6)$ and $F(5,5)$, and the center of the circle lies on the line $l:3x-5y+9=0$.
1. Find the equation of the circle $C$.
2. The line passing through the point $M(0,3)$ intersects the circle $C$ at points $A$ and $B$. The question is: Does there exist a fixed point $N$ on the line $y=3$ such that $k_{AN}=-k_{BN}$ (where $k_{AN}$ and $k_{BN}$ are the slopes of the lines $AN$ and $BN$ respectively)? If it exists, find the coordinates of point $N$; if not, explain the reason.

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