A $20$ lb$.$ monkey is attached to a $50$ ft chain that weighs $0\mathrm{.}5$ lb$/$ft$.$ The other end of the chain is attached to the $40-$ft$-$high ceiling of the monkey$$s cage. Find the amount of work the monkey does by climbing up her chain to the ceiling. $($Hint$.$ Split this calculation up into three parts: $($i$)$ the monkey, $($ii$)$ the $10$ ft of chain initially on the ground, $($iii$)$ the $15$ ft of chain initially hanging that will move up$.)$