Two light beams fall on a transparent material block at point \( 1 \) and \( 2 \) with angle \( \theta_{1} \) and \( \theta_{2} \), respectively, as shown in figure. After refraction, the beams intersect at point \( 3 \) which is exactly on the interface at other end of the block.
Given: the distance between \( 1 \) and \( 2 \), \( d = 4\sqrt{3} \, \text{cm} \) and \( \theta_{1} = \theta_{2} = \cos^{-1}\!\left(\frac{n_{2}}{2 n_{1}}\right) \), where refractive index of the block \( n_{2} > \) refractive index of the outside medium \( n_{1} \), then the thickness of the block is __________ cm.
