In a right triangle $ABC$, let $D$ be the midpoint of the hypotenuse $AB$. The circumradius of the triangle is $\frac{5}{2}$ and $|BC| = 3$. Find the distance between the circumcenter of $\triangle ACD$ and the incenter of $\triangle BCD$. The 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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