
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 $C: x^2+y^2-4x-6y+12=0$ with its center at point $C$, and point $A(3,5)$, find:
$(1)$ The equation of the tangent line to the circle passing through point $A$;
$(2)$ If point $O$ is the origin, and segment $OA$ and $OC$ are connected, calculate the area $S$ of the triangle $\triangle AOC$.

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