Consider a configuration of $$n$$ identical units, each consisting of three layers. The first layer is a column of air of height $$h = \dfrac{1}{3}$$ cm, and the second and third layers are of equal thickness $$d = \dfrac{\sqrt{3}-1}{2}$$ cm, and refractive indices $$\mu_1 = \dfrac{\sqrt{3}}{2}$$ and $$\mu_2 = \sqrt{3}$$, respectively. A light source O is placed on the top of the first unit as shown in the figure. A Ray of light from $$O$$ is incident on the second layer of the first unit at an angle of $$\theta = 60^\circ$$ to the normal. For a specific value of $$n$$, the ray of light emerges from the bottom of the configuration at a distance $$l = \dfrac{8}{\sqrt{3}}$$ cm, as shown in the figure. The value of $$n$$ is _______.