In the triangle $ABC$, it is known that $AC <AB$. Let $\ell$ be tangent to the circumcircle of triangle $ABC$ drawn at point $A$. A circle with center $A$ and radius $AC$ intersects segment $AB$ at point $D$, and line $\ell$ at points $E$ and $F$. Prove that one of the lines $DE$ and $DF$ passes through the center inscribed circle of triangle $ABC$.