Advanced Number Theory · Challenge problem

A circle of radius $1$ is tangent to both sides of a right angle. A second, smaller circle is tangent to both sides and to the first circle; a third is tangent to both sides and to the second; and so on forever, the circles shrinking toward the vertex. The total area of all the circles is $\dfrac{(a\sqrt b + c)\pi}{d}$, where $a, b, c, d$ are positive integers, $b$ is squarefree, and $a$, $c$, $d$ have no common factor. Find $a + b + c + d$.