
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 hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with a focal length of $4$ and an imaginary axis length of $2$, where the left and right foci are $F_{1}$ and $F_{2}$ respectively. The line $l: y = kx + m\ (m \neq 0)$ intersects the curve $C$ at two distinct points $A$ and $B$.  
$(1)$ Find the equation of the hyperbola $C$ and its eccentricity $e$.  
$(2)$ If the line $l$ passes through the point $F_{2}$ and $|AB| = 2\sqrt{3}$, find the equation of the line $l$.  
$(3)$ Does there exist a line $l$ such that both points $A$ and $B$ lie on a circle with center $D(0, -1)$? If yes, find the range of values for $m$; if not, explain why.

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