2025 AMC 10B Question 20

Four congruent semicircles are inscribed in a square of side length $1$ so that their diameters are on the sides of the square, one endpoint of each diameter is at a vertex of the square, and adjacent semicircles are tangent to each other. A small circle centered at the center of the square is tangent to each of the four semicircles, as shown below.



The diameter of the small circle can be written as $(\sqrt{a} + b)(\sqrt{c} + d),$ where $a,$ $b,$ $c,$ and $d$ are integers. What is $a + b + c + d?$