
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
Determine the locus equation of vertex $A$ in triangle $\triangle ABC$, given that $B(-5,0)$ and $C(5,0)$ are vertices of the triangle and $\sin \angle ABC - \sin \angle ACB = \frac{3}{5} \sin \angle BAC$.

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