Solving for a Length given the Medians

Geometry Level pending

Consider a \(\triangle ABC\) with midpoints \(D\), \(E\), and \(F\) on sides \(AB\), \(BC\), and \(CA\) respectively. The lengths of segments \(AE\), \(BF\), and \(CD\) are \(4\), \(\sqrt{10}\), and \(\sqrt{3}\) respectively. The length of segment \(EF\) can be written in the form \(\dfrac{j}{k}\) where \(j\) and \(k\) are coprime positive integers. Calculate \(j + k\).

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