Three concentric circles are shown. The two largest circles have radii of $12$ and $13.$ If the area of the ring between the two largest circles equals the area of the smallest circle, determine the radius of the smallest circle.

[asy]
import graph;
filldraw(circle((0,0),12), lightgray, black+linewidth(1));
filldraw(circle((0,0),10), white, black+linewidth(1));
filldraw(circle((0,0),6), gray, black+linewidth(1));
dot((0,0));
[/asy]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Three concentric circles are shown. The two largest circles have radii of $12$ and $13.$ If the area of the ring between the two largest circles equals the area of the smallest circle, determine the radius of the smallest circle.

[asy]
import graph;
filldraw(circle((0,0),12), lightgray, black+linewidth(1));
filldraw(circle((0,0),10), white, black+linewidth(1));
filldraw(circle((0,0),6), gray, black+linewidth(1));
dot((0,0));
[/asy]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
5
Three concentric circles are shown. The two largest circles have radii of $12$ and $13.$ If the area of the ring between the two largest circles equals the area of the smallest circle, determine the radius of the smallest circle.

[asy]
import graph;
filldraw(circle((0,0),12), lightgray, black+linewidth(1));
filldraw(circle((0,0),10), white, black+linewidth(1));
filldraw(circle((0,0),6), gray, black+linewidth(1));
dot((0,0));
[/asy]
train