Semicircles with centers $A,B$ and radii $3,2$ share bases with a large semicircle $JK$, are externally tangent to each other and internally tangent to the large semicircle. A circle centered at $P$ is externally tangent to the two smaller semicircles and internally tangent to the largest. What is the radius of the circle at $P$?

1. With radius $x$: $PA=x+3,PB=x+2,PO=5-x,AO=2,OB=3$; the law of cosines on $\triangle PAO,\triangle POB$ (eliminating the shared supplementary angle) gives $2(x+2)^2+3(x+3)^2=30+5(5-x)^2$.
2. Solving: $x=\dfrac{30}{19}$.
3. The answer is $\boxed{\dfrac{30}{19}}$.