Minimize The Ratio

Round 19

Statement

    In $\triangle ABC$, let $h_a, h_b, h_c$ be the altitudes from $A,B,C$ and $r$ be it's inradius. The minimum possible value of $\dfrac{h_a+h_b+h_c}{r}$ can be written as $\dfrac{p}{q}$ where $p$ and $q$ are positive coprime integets. Determine $p+q$.
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