
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 ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{2} = 1$ with foci $F_1$ and $F_2$, and a point $P$ on the ellipse such that $|PF_1| = 4$, find the cosine value of $\angle F_1 P F_2$.

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