
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 the Cartesian coordinate system \(xOy\) the vertices of \(\Delta ABC\) are \(A(0,4)\) and \(C(0,-4)\), and vertex \(B\) is on the ellipse \(\dfrac{x^2}{9}+\dfrac{y^2}{25}=1\), calculate \(\dfrac{\sin (A+C)}{\sin A+\sin C}\).

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