Let $△A_0{B}_0{C}_0$ be a triangle whose angle measures are exactly $59.9^{\circ }$, $60^{\circ }$, and $60.1^{\circ }$. 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 an angle of $△A_n{B}_n{C}_n$ is less than $1^{\circ }$?