
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
The line $x-y=0$ intersects the circle $C:\left(x-1\right)^{2}+\left(y-3\right)^{2}=9$ at points $A$ and $B$. <br/>$(1)$ Find the length of the chord $|AB|$; <br/>$(2)$ Find the equation of the line passing through point $P\left(4,-3\right)$ and tangent to circle $C$.

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