On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that $$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$