A Constant Focal Angle for an Escribed Tangent Triangle

As shown in the figure, from a point $P$ outside an ellipse, the two tangent lines $PA$ and $PB$ are drawn, touching the ellipse at $A$ and $B$. The tangent line to the ellipse at a further point $C$ meets $PA$ at $M$ and $PB$ at $N$, so that the ellipse is escribed to triangle $PMN$: it touches side $MN$ and the extensions of sides $PM$ and $PN$. Let $F$ be a focus of the ellipse.

Prove that the angle $\angle MFN$ subtended by $MN$ at $F$ does not depend on the position of the point $C$.