
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 $(2,4)$ and $(1,3)$, and the center of circle $C$ is on the line $x-y+1=0$. A line $l$ with slope $k$ passes through point $A(0,1)$ and intersects circle $C$ at points $M$ and $N$.

$(1)$ Find the equation of circle $C$;

$(2)$ If $O$ is the origin, and $\overrightarrow{OM} \cdot \overrightarrow{ON}=12$, find the equation of line $l$.

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