Let $\triangle{A_0B_0C_0}$ be a triangle whose angle measures are exactly $59.999^{\circ}$, $60^{\circ}$, and $60.001^{\circ}$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$& to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle{A_nB_nC_n}$ is obtuse?