
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 the line $l$ with its symmetric point being $P'(b + 1, a - 1)$, find the equation of the circle $C'$ which is symmetric to the circle $C: x^2 + y^2 - 6x - 2y = 0$ with respect to the line $l$.

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