
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
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5,$ as shown in the diagram. Find $r.$
[asy]
size(7cm);
draw(scale(sqrt(5), sqrt(5)/2)* unitcircle);
draw((0,-1.5)--(0,1.7),EndArrow);
draw((-3,0)--(3,0),EndArrow);
draw(Circle( (sqrt(1.25),0), sqrt(1.25) ));
draw(Circle( (-sqrt(1.25),0), sqrt(1.25) ));
label("$x$",(3,0),E);label("$y$",(0,1.7),N);
[/asy]

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