
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 $F_{1}(-4,0)$ and $F_{2}(4,0)$ are the two foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, and point $P$ is on the ellipse such that the area of $\triangle PF_{1}F_{2}$ is $3 \sqrt{3}$, find the value of $\cos ∠F_{1}PF_{2}$.

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