Let $ABC$ be an isosceles triangle with $|AB| = |AC| = 10$ and $|BC| = 12$. Points $P$ and $R$ are on $[BC]$ such that $|BP| = |RC| = 3$. Points $S$ and $T$ are the midpoints of $[AB]$ and $[AC]$, respectively. If $M$ and $N$ are the feet of the perpendiculars from $S$ and $R$ to $PT$, find the length of $|MN|$. The original answer is in the form \frac{k\sqrt{m}}{n}, please provide the value of k + m + n.

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