
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 triangle $\triangle ABC$ with vertices $A$ and $B$ located at the left and right foci of the ellipse $x^2+5y^2=5$, respectively, and the internal angles $A$, $B$, $C$ satisfy the relation $\sin B - \sin A = \sin C$.
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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