Draw a plane $\pi$ through $M$ that passes through the line $l$ connecting centers of the spheres. Let it intersect the respective spheres with circles $\omega_1, \omega_2$ and intersect the circles $ABC, A_1B_1C_1$ at points $X, Y$ and $X_1, Y_1$ respectively. Because the circles $ABC$ and $A_1B_1C_1$ are inverse with respect to $M$, the points $M, X, X_1$ lie on a line and so do $M, Y, Y_1$. Note that $\varphi =\angle (YX_1, YY_1)= \angle (X_1Y, X_1X)+ \angle (MX, MY)$ remains constant while $M$ slides over the circle $S_2 \cap \pi$ thus the distance from the center $O_2$ of the sphere $S_2$ to the line $X_1Y_1$ is equal to $R_2 \cos \varphi$ and remains constant where $R_2$ stands for the radius of $S_2$. It follows that while $M$ slides over the circle $S_2\cap \pi$ the lines $X_1Y_1$ and therefore the planes $A_1B_1C_1$ too remain tangent to the circle $\gamma$ with center $O_2$ and radius $R_2 \cos \varphi$. It's not hard to see that when $\pi$ rotates around $l$ $\varphi$ remains constant thus the sphere $\Gamma$ formed by the rotation of $\gamma$ around $l$ is tangent to all planes $A_1B_1C_1$.