Value Plugging May Not Help ...

Algebra Level 5

The sum of all values of $x$ that satisfy

$\large{\sqrt [ 3 ]{ 6(5x+6) }-\sqrt [ 3 ]{ 5(6x-11) }=1},$

can be expressed as $\dfrac{m}{n},$ where $m$ and $n$ are coprime integers. Find $m+n.$