
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 the line segment $AB$ with endpoint $B$ at coordinates $(0,3)$, and endpoint $A$ moves on the circle $C$: $(x+1)^{2}+y^{2}=4$.
$(1)$ Find the equation of the trajectory of the midpoint $M$ of line segment $AB$;
$(2)$ A line $l$ passing through point $B$ intersects the circle $C$ at two points $A$ and $B$, and the length of chord $AB$ is $\dfrac{2\sqrt{19}}{5}$. Find the equation of line $l$.

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