Symmetric sum of altitudes

Geometry Level 3

Triangle ABCABC has area 15 and perimeter 20. Furthermore, the product of the 3 side lengths is 255. If the three altitudes of the triangle have lengths d,ed, e, and ff, then the value of de+ef+fdde+ef+fd can be written as mn\frac{m}{n} for relatively prime positive integers mm and nn. What is m+nm+n?

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