
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 fixed point $A(1,-2)$, point $B$ is a moving point on the circle $(x+1)^{2} + (y+4)^{2} = 4$.  
$(1)$ Find the equation of the locus of the midpoint $C$ of $AB$.  
$(2)$ If the line $l$ passing through the fixed point $P(\frac{1}{2},-2)$ intersects the locus of $C$ at points $M$ and $N$, and $|MN| = \sqrt{3}$, find the equation of the line $l$.

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