
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 a hyperbola, $F_1(-\sqrt{5},0)$ and $F_2(\sqrt{5},0)$, and a point $P$ on the hyperbola such that the dot product of vectors $\overrightarrow{PF_1}$ and $\overrightarrow{PF_2}$ is zero, and the product of their magnitudes $|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}| = 2$, find the standard equation of the hyperbola.

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