A student is calibrating a thin film deposition system with interference by shining a $633$ nm laser on a film and measuring the reflection. So that the incident and reflected beams do not overlap, the laser is held at a slight angle $($but the angle can be ignored in the questions below$).$ The systems makes a thin film of cryolite $($n $=1\mathrm{.}34)$ on a sapphire $($n $=1\mathrm{.}77)$ substrate. For questions $1$ and $2$ below, make sure to show your work by drawing a diagram of the incident and reflected rays and by calculating the phase difference.

$(1)$ Based on the deposition settings, the film made should be between $100$ and $300$ nm thick to produce maximal reflections. What is the film's precise thickness?

$(2)$ A different film is made with a thickness between $350$ nm and $400$ nm with other settings. This film has zero reflection. What is the film's precise thickness?

$(3)$ Finally, a third film is made that is between $50$ nm and $100$ nm thick. The reflected light has half of the amplitude of the constructive reflection in question #$1.$ What is this film's precise thickness?