Iterated Pedal Triangles

Triangle $A_0B_0C_0$ has interior angles $59.999^\circ$, $60^\circ$, $60.001^\circ$. For each positive integer $n$, let $A_n$ be the foot of the perpendicular from $A_{n-1}$ to side $B_{n-1}C_{n-1}$, and define $B_n$ and $C_n$ analogously. Find the least positive integer $n$ for which triangle $A_nB_nC_n$ is obtuse.