
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 both foci of an ellipse. The ellipse has the equation $\frac{x^2}{9} + \frac{y^2}{4} = 1$. The circle passes through exactly four points on the ellipse. The range for all possible values of the circle's radius $r$ forms an interval $[a, b)$. Determine $a+b$.
A) $\sqrt{5} + 8$
B) $\sqrt{5} + 19$
C) $\sqrt{5} + 8.5$
D) $\sqrt{5} + 9$

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