Greatest Integer Function

Algebra Level 5

$x$ is a real number and satisfies the equation:
$\frac{1}{[x]}+\frac{1}{[2x]}=x-[x]+\frac{1}{3}$ .
The sum of all such numbers can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$

Details and Assumptions

$[x]$ denotes the Greatest Integer Function: $[x]=n$ , where $n$ is the greatest integer less than $x$ .