SHARYGIN 2022 Problem 24

Let $OABCDEF$ be a hexagonal pyramid with base $ABCDEF$ circumscribed around a sphere $\omega$ . The plane passing through the touching points of $\omega$ with faces $OFA$ , $OAB$ and $ABCDEF$ meets $OA$ at point $A_1$ , points $B_1$ , $C_1$ , $D_1$ , $E_1$ and $F_1$ are defined similarly. Let $\ell$ , $m$ and $n$ be the lines $A_1D_1$ , $B_1E_1$ and $C_1F_1$ respectively. It is known that $\ell$ and $m$ are coplanar, also $m$ and $n$ are coplanar. Prove that $\ell$ and $n$ are coplanar.