
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 $P$ is a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\frac{{\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}}}{{|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}|}}=\frac{1}{2}$, then the area of $\triangle F_{1}PF_{2}$ is ______.

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