
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 $F_1$ and $F_2$ are the foci of the hyperbola $${x^2 - y^2 = 1}$$, and a circle with $F_1F_2$ as the diameter intersects with one asymptote at points $P$ and $Q$. Determine the area of triangle $F_1PQ$.

A) $$\frac{\sqrt{2}}{2}$$
B) 1
C) $$\sqrt{2}$$
D) 2

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