
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 the parabola $y^2=4x$ and the ellipse $\frac{x^{2}}{9}+ \frac{y^{2}}{k}=1$ share a common focus $F_1$, the other focus of the ellipse is $F_2$, and $P$ is one of the intersection points of these two curves. Calculate the area of $\triangle PF_1F_2$.

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