
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 curve $C$ is the locus of points whose distances to two fixed points $O(0,0)$ and $A(3,0)$ have a ratio of $\dfrac{1}{2}$.  
$(1)$ Find the equation of curve $C$ and identify the type of curve;  
$(2)$ A line $l$ passing through $(-2,2)$ intersects curve $C$ at points $M$ and $N$, with $|MN|=2\sqrt{3}$, find the equation of line $l$.

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