A particle moves along the curve $y = \frac{3(x+6)}{x+4}$, where $x \ne -4$, in such a way that the $y$-coordinate of the particle is increasing at a constant rate of $\frac{4}{27}$ units per second. Find the $x$-coordinates of the particle at the instant that the $x$-coordinate of the particle is decreasing at a rate of 2 units per second.