Collinearity of Points in Triangle with Angle Bisectors

Let triangle $ABC$ have angle bisectors $BB_1$ and $CC_1$ that intersect at point $M$. The angle bisectors $B_1B_2$ and $C_1C_2$ of triangle $AB_1C_1$ intersect at point $N$. Prove that points $A$, $M$, and $N$ are collinear.