
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
Let $F\_1$ and $F\_2$ be the foci of the ellipse $C\_1$: $\frac{x^2}{6} + \frac{y^2}{2} = 1$, and $P$ be an intersection point of the hyperbola $C\_2$: $\frac{x^2}{3} - y^2 = 1$ and $C\_1$. Determine the area of $\triangle PF\_1F\_2$.

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