
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 a circle with center $A(-1,2)$ intersecting with a dynamic line $l$ passing through point $B(-2,0)$, intersecting the circle at points $M$ and $N$, and tangent to the line $x+2y+7=0$; circle $\left(x-3\right)^{2}+y^{2}=20$ is symmetric with respect to the line $2x-y-1=0$; the length of the common tangent of circle $\left(x-3\right)^{2}+\left(y-2\right)^{2}=5$ is $\sqrt{11}$. Choose any one of these three conditions and complete the blanks in the above question to answer the following questions:
$(1)$ Find the equation of circle $A$;
$(2)$ When $|MN|=2\sqrt{19}$, find the equation of line $l$.

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