
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 a point $P$ on the hyperbola $\dfrac{x^2}{2} - y^2 = 1$, with $F_1$ and $F_2$ as the two foci of the hyperbola, if $|PF_1| + |PF_2| = 4\sqrt{2}$, find the area of $\triangle PF_1F_2$.

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