
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
In $\triangle ABC$, the sides opposite to vertices $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that $B(-1,0)$, $C(1,0)$, and $b$, $a$, $c$ form an arithmetic sequence.

$(1)$ Find the equation of the trajectory of vertex $A$;

$(2)$ Suppose the trajectory of vertex $A$ intersects with the line $y = kx + m$ at two distinct points $M$ and $N$. If there exists a line $l$ passing through point $P(0,-\frac{1}{2})$ such that points $M$ and $N$ are symmetric with respect to $l$, find the range of the real number $m$.

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