Problem $\backslash$#$4(10$ points$)$

Steam at constant pipeline conditions of $300$ kPa and $\backslash (250^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ passes slowly through a valve into a well$-$insulated, frictionless, and initially evacuated piston$-$cylinder assembly. A pressure of $100$ kPa in the cylinder is needed to support the piston due to its own weight and the ambient atmosphere. The volume under the piston is $\backslash (5\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ with $0\mathrm{.}5$ kg of water inside the cylinder at the end of the process.

Find: $($a$)$ the boundary work done during this process, $($b$)$ the final temperature of the water in the cylinder, and $($c$)$ the phase of water in the cylinder at the final state $($please calculate quality if it is a saturated mixture$).$