
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 that the point $P(a,b)$ is symmetric to line $l$ at point $P'(b+1,a-1)$, determine the equation of the circle $C'$ that is symmetric to line $l$ for circle $C$: $x^{2}+y^{2}-6x-2y=0$. Also, find the length of the common chord of circles $C$ and $C'$.

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