
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 \( B(-6,0) \) and \( C(6,0) \) are two vertices of triangle \( \triangle ABC \), and the interior angles \( \angle A, \angle B, \angle C \) satisfy \( \sin B - \sin C = \frac{1}{2} \sin A \). Find the equation of the locus of vertex \( A \).

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