Given are two concentric circles $\Omega$ and $\omega$. Each of the circles $b_1$ and $b_2$ is externally tangent to $\omega$ and internally tangent to $\Omega$, and $\omega$ each of the circles $c_1$ and $c_2$ is internally tangent to both $\Omega$ and $\omega$. Mark each point where one of the circles $b_1, b_2$ intersects one of the circles $c_1$ and $c_2$. Prove that there exist two circles distinct from $b_1, b_2, c_1, c_2$ which contain all $8$ marked points. (Some of these new circles may appear to be lines.)