
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 the circle $C: x^{2}+y^{2}-4x-6y+12=0$, and the point $A(3,5)$, find:

$(1)$ The equation of the tangent line to the circle passing through point $A$;

$(2)$ Let $O$ be the origin of the coordinate system, connect $OA$ and $OC$, calculate the area $S$ of $\triangle AOC$.

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