
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
An ellipse has two foci at $F\_1(-4,0)$ and $F\_2(4,0)$. Point $P$ lies on the ellipse, and if $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = 0$ and the area of $\triangle PF\_1F\_2$ is $9$, what is the standard equation of the ellipse?

A) $\frac{x^2}{25} + \frac{y^2}{9} = 1$
B) $\frac{x^2}{32} + \frac{y^2}{16} = 1$
C) $\frac{x^2}{36} + \frac{y^2}{20} = 1$
D) $\frac{x^2}{20} + \frac{y^2}{4} = 1$

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