YDSE: Film, Medium & Two Wavelengths question from JEE Main 2023 (1 Feb, Shift 2)

As shown in the figure, in Young's double slit experiment, a thin plate of thickness $t = 10\ \mu\text{m}$ and refractive index $\mu = 1.2$ is inserted in front of slit $S_1$. The experiment is conducted in air $(\mu = 1)$ and uses a monochromatic light of wavelength $\lambda = 500\text{ nm}$. Due to the insertion of the plate, central maxima is shifted by a distance of $x\beta_0.\ \beta_0$ is the fringe-width before the insertion of the plate. The value of the $x$ is _____________.