
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
A circle passes through one vertex and one focus of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, and its center lies on the hyperbola. Find the distance from the center of the circle to the center of the hyperbola.

A) 4
B) $\frac{16}{3}$
C) $\frac{4\sqrt{7}}{3}$
D) 5

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