
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
It is known that a hyperbola has its center at the origin, with two foci F<sub>1</sub> and F<sub>2</sub> located at ($$\sqrt{5}$$, 0) and ($$-\sqrt{5}$$, 0) respectively. A point P on the hyperbola is such that PF<sub>1</sub> is perpendicular to PF<sub>2</sub> and the area of triangle PF<sub>1</sub>F<sub>2</sub> is 1. Find the equation of the hyperbola.

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