
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 passes through the point (4,0), and its asymptote equations are $y = \pm x$. If a circle C goes through one vertex and one focus of the hyperbola, and the center of the circle lies on the hyperbola itself, find the distance from the center of the circle to the center of the hyperbola.

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