Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$
[asy]
size(7cm);
draw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);
draw((0,-1.5)--(0,1.7),EndArrow);
draw((-3,0)--(3,0),EndArrow);
draw(Circle( (sqrt(0.96),0), sqrt(0.96) ));
draw(Circle( (-sqrt(0.96),0), sqrt(0.96) ));
label("$x$",(3,0),E);label("$y$",(0,1.7),N);
[/asy]
user