A rod of length \( 2 \, \mathrm{cm} \) makes an angle \( \dfrac{2\pi}{3} \) rad with the principal axis of a thin convex lens. The lens has a focal length of \( 10 \, \mathrm{cm} \) and is placed at a distance of \( \dfrac{40}{3} \, \mathrm{cm} \) from the object as shown in the figure. The height of the image is \( \dfrac{30\sqrt{3}}{13} \, \mathrm{cm} \) and the angle made by it with respect to the principal axis is \( \alpha \) rad. The value of \( \alpha \) is \( \dfrac{\pi}{n} \) rad, where \( n \) is _______.
