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 $a + b + c + d$? $\textbf{(A) } 3 \qquad$ $ \textbf{(B) } 5 \qquad$ $ \textbf{(C) } 8 \qquad$ $ \textbf{(D) } 9 \qquad$ $ \textbf{(E) } 11$