
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 $A(-1,2)$ tangent to the line $l_{1}: x+2y+7=0$, and a moving line $l$ passing through point $B(-2,0)$ intersects the circle with center $A$ at points $M$ and $N$. <br/>$(1)$ Find the equation of circle $A$; <br/>$(2)$ When $|MN|=2\sqrt{19}$, find the equation of line $l$.

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