
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 the two foci of the hyperbola, $F_1(-\sqrt{5},0)$ and $F_2(\sqrt{5},0)$, and a point $P$ on the hyperbola, with $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$ and $|PF_1| \cdot |PF_2| = 2$, find the standard equation of the hyperbola.

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