
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 circles $C_{1}$: $x^{2}+y^{2}-2x-6y-1=0$ and $C_{2}$: $x^{2}+y^{2}-10x-12y+45=0$.<br/>$(1)$ Find the equation of the common chord of circles $C_{1}$ and $C_{2}$ and the length of the common chord;<br/>$(2)$ Find the equation of the line passing through point $P(9,1)$ and tangent to circle $C_{2}$.

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