
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 of circle $C$ lies on the line $y=x+1$, and circle $C$ is tangent to the $x$-axis. Point $P(-5,-2)$ lies on circle $C$, and point $Q(-4,-5)$ lies outside circle $C$. <br/>$(1)$ Find the equation of circle $C$; <br/>$(2)$ If the line $l$ passing through the point $(-2,-4)$ intersects circle $C$ at points $A$ and $B$ such that $|AB|=2\sqrt{3}$, find the equation of line $l$.

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