
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 points $M(0,-5)$, $N(0,5)$, and a circle passing through these points with center $P$, find the equation of the locus of the vertex $P$.

A) $\frac{x^2}{169}+\frac{y^2}{144}=1 (y \neq 0)$

B) $\frac{y^2}{169}+\frac{x^2}{144}=1 (x \neq 0)$

C) $\frac{x^2}{169}+\frac{y^2}{25}=1 (y \neq 0)$

D) $\frac{y^2}{169}+\frac{x^2}{25}=1 (x \neq 0)$

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