A heat engine operates between two reservoirs at T$2=600$KT$\_2=600\backslash ,\backslash$text$\{$K$\}$T$2=600$K and T$1=350$KT$\_1=350\backslash ,\backslash$text$\{$K$\}$T$1=350$K$.$ It takes in QH$=1\mathrm{.}00103$JQ$\_$H $=1\mathrm{.}00\backslash$times $10^3\backslash ,\backslash$text$\{$J$\}$QH$=1\mathrm{.}00103$J of energy from the higher$-$temperature reservoir and performs W$=250$JW $=250\backslash ,\backslash$text$\{$J$\}$W$=250$J of work.

$($a$)$ Find the entropy change of the Universe $($SU$\backslash$Delta S$\_$U$$SU$).$

$($b$)$ Calculate the work $($WWW$)$ that could have been done by an ideal Carnot engine operating between these two reservoirs.