
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
Consider the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{2} = 1$ and the hyperbola $\frac{x^{2}}{3} - y^{2} = 1$ with common foci $F_{1}$ and $F_{2}$. Let $P$ be one of the intersection points of the two curves. Find the value of $\cos \angle F_{1}PF_{2}$.

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