
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 that in $\triangle ABC$, the two vertices $A$ and $B$ are respectively the left and right foci of the ellipse $x^{2}+5y^{2}=5$, and the three internal angles $A$, $B$, $C$ satisfy the relation $\sin B - \sin A = \frac{1}{2}\sin C$.

$(1)$ Find the length of segment $AB$;

$(2)$ Find the equation of the trajectory of vertex $C$.

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