Tricky Triplet Trepidation

Algebra Level 3

Let \((a, b, c)\) be an ordered triplet of positive real numbers such that

\[ \dfrac{3a + 8b + 36c}{\sqrt{a^2 + 4b^2 + 9c^2}} = 13. \]

The sum of all possible values of \(\dfrac{a + b}{c}\) can be written in the form \(\dfrac{p}{q},\) where \(p\) and \(q\) are positive coprime integers. Find the value of \(p + q.\)

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