
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
(1) Given that the major axis of an ellipse is 10, and the foci are $F\_1(-4,0)$ and $F\_2(4,0)$, find the standard equation of the ellipse.

(2) In the rectangular coordinate system $xOy$, the left and right foci of the ellipse $C: \frac{x^2}{25} + \frac{y^2}{9} = 1$ are $F\_1$ and $F\_2$, respectively. $P$ is a point on the ellipse $C$ such that $PF\_1 \perp PF\_2$. Find the area of $\triangle PF\_1F\_2$.

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