$(1)$ A piston$-$cylinder device initially contains $0\mathrm{.}5$ kg of steam at $4$ MPa and $400C.$ Now the steam loses heat to the surroundings and the piston moves down, hitting a set of stops at which point the cylinder contains mixture with the quality of $0\mathrm{.}25.$ The cooling continues until the cylinder contains water at $220C.$ Determine $($a$)$ the final pressure and quality $($if mixture$),($b$)$ the boundary work, $($c$)$ the amount of heat transfer when the piston first hits the stops, and $($d$)$ the total heat transfer. $(25\%)$