
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
The graph of hyperbola $xy = 1$ is given. Convert this into a rotated coordinate system where axes are aligned with the asymptotes of the hyperbola, calculate the coordinates of the foci in the new system, and find the distance between these foci.

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