In triangle $ABC$, $AB = 52$, $BC = 34$, and $CA = 50$. The side $BC$ is divided into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. What is the smallest possible value of $n$?

Let's think step by step and output the final answer within \boxed{}.
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