
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 $F_1$ and $F_2$ are two foci of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, $P$ is a point on the ellipse $C$ and $PF_1 \perp PF_2$. If the area of $\triangle PF_1F_2$ is $9$, and the perimeter of $\triangle PF_1F_2$ is $18$, find the equation of the ellipse.

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