Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$. Let $\omega$ be the circle that touches the sides $AB$, $BC$, and $AD$. A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$. The ratio $\frac{AK}{KD}$ can be written as $\frac{a}{b}$. Find the value of $a+b$