KöMaL problem A, 2018

Let $P$ be a point in the plane of triangle $ABC$. Denote the reflections of $A$, $B$, $C$ about $P$ by $A'$, $B'$ and $C'$, respectively. Let $A''$, $B''$, $C''$ be the reflections of $A'$, $B'$, $C'$ over the lines $BC$, $CA$ and $AB$, respectively. Let the line $A''B''$ intersect $AC$ at $A_b$ and let $A''C''$ intersect $AB$ at a point $A_c$. Denote by $\omega_A$ the circle through the points $A$, $A_b$, $A_c$. The circles $\omega_B$, $\omega_C$ are defined similarly. Prove that $\omega_A$, $\omega_B$, $\omega_C$ are coaxial, i.e., they share a common radical axis.