Minimum Value of Cotangent in Triangle?

Geometry Level pending

In the triangle ABC, the medians from point B and point C are perpendicular to each other. If \(\frac{a}{b}\) is the minimum value of \(\cot \angle B + \cot \angle C\), where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b\).

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