Question 85
Let $ABCD$ be a square. Points $E$ and $F$ lie on sides $CD$ and $DA$, respectively, such that $\overline{EBF} = 45^\circ$. Given that $DE = 12^{34}$, determine the number of triples $(k, m, n)$ of positive integers with $AB = k$, $DF = m$, and $EF = n$.