
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
Consider an ellipse with the equation \(4x^2 + 9y^2 = 36\). A circle of radius \(r\) passes through both foci of the ellipse and exactly six points on the ellipse. Find the set of all possible values of \(r\) and express this set as an interval \([a, b)\). What is \(a+b\)?
A) 12
B) 11
C) 15
D) 13
E) 10

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