
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 center $C$ of circle $C$ lies on the line $x-y-1=0$, and the circle is tangent to the line $2x+3y-10=0$ at point $P(2,2)$.<br/>$(1)$ Find the equation of circle $C$;<br/>$(2)$ If the chord $AB$ cut off by line $l$ passing through point $Q(2,3)$ on circle $C$ has a length of $6$, find the equation of line $l$.

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