Q$3$ ABC company is considering replacing an old machine with a new one. The old machine was purchased $6$ years ago for USD $120,000$ and has a remaining useful life of $2$ years. The new machine costs USD $250,000$ and has an expected useful life of $8$ years. The annual operating cost of the old machine and new machine are USD $40,000$ and USD $30,000$ respectively. The salvage value of the old machine and new machine are estimated to be USD $20,000$ and USD $100,000$ respectively. The company's cost of capital is $\backslash (8\backslash \%\backslash ).$

$($a$)$ Analyze the economic service life $($ESL$)$ of the old machine.

$($b$)$ Analyze the ESL of the new machine.

$($c$)$ Analyze the annual worth $($AW$)$ of the old machine.

$($d$)$ Analyze the AW of the new machine.

$($e$)$ Should the company replace the old machine with the new one? Justify your answer based on the AW and ESL analysis.