
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 point $P(3,4)$ is on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $F_1$, $F_2$ are the two foci of the ellipse. If $PF_1 \perp PF_2$, find:

(1) The equation of the ellipse;

(2) The area of $\triangle PF_1F_2$.

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