Geometry Problem

In the diagram, four circles of radius 1 with centres $P, Q, R,$ and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown.

The radius of the circle with center $R$ is decreased so that

the circle with center $R$ remains tangent to $BC$,

the circle with center $R$ remains tangent to the other three circles, and

the circle with center $P$ becomes tangent to the other three circles.

The radii and tangencies of the other three circles stay the same. This changes the size and shape of $\triangle ABC$.  $r$ is the new radius of the circle with center $R$. $r$ is of the form $\frac{a+\sqrt{b}}{c}$. Find $a+b+c$.