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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