HMMT 2020 G10

Let $\Gamma$ be a circle of radius 1 centered at $O$. A circle $\Omega$ is said to be friendly if there exist distinct circles $\omega_{1}, \omega_{2}, \ldots, \omega_{2020}$, such that for all $1 \leq i \leq 2020, \omega_{i}$ is tangent to $\Gamma, \Omega$, and $\omega_{i+1}$. (Here, $\omega_{2021}=\omega_{1}$.) For each point $P$ in the plane, let $f(P)$ denote the sum of the areas of all friendly circles centered at $P$. If $A$ and $B$ are points such that $O A=\frac{1}{2}$ and $O B=\frac{1}{3}$, determine $f(A)-f(B)$.