
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 point $P(4,0)$, the center of circle $C$ lies on the line $x-y-4=0$, and circle $C$ tangents the $y$-axis at point $M(0,-2)$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If a line $l$ passes through point $P$ and intersects circle $C$ with a chord length of $2\sqrt{2}$, find the equation of line $l$.

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