Godfrey has just recovered from a serious illness and is planning to start his early retirement at the age of $55($i$.$e$.,$ start the retirement right after his $55$th birthday$).$ According to Godfreys calculations, he would need an average of $$40,000$ per month for the living expenses during the retirement years. The monthly withdrawal will be made at the beginning of each $($retirement$)$ month. Godfrey would also like to leave his teenage daughter $$20$m which she would be able to inherit in $10$ years after his retirement. The retirement plan Godfrey purchases will provide a fixed interest return of $12\%,$ compounded monthly. a$)$ How much money $($the target amount$)$ will Godfrey need to have in his retirement savings account at his $55$th birthday so that he meet all expected expenses if he expects a $15-$year retirement life? $(5$ marks$)$ To meet the target amount, the retirement plan provides Godfrey with $2$ choices$/$plans: Plan $(1)$: make a single deposit of $$4,086,356\mathrm{.}21$ into the retirement plan account $($that earns a fixed interest return of $12\%,$ compounded monthly, as stated above$)$; or Plan $(2)$: make $72$ month$-$end deposits of $$50,000$ each into the retirement plan account $($the last of which will be made exactly at his $55$th birthday$)$ PLUS one single payment at his $55$th birthday. b$)$ If Godfrey chooses Plan $(1),$ when at the latest $($in terms of age$)$ will be required to make the lump$-$sum deposit in order to have enough money to meet the retirement expenses? $(4$ marks$)$ c$)$ If Godfrey chooses Plan $(2),$ what is the size of the single payment that he will be required to make at his $55$th birthday in order to have enough money to meet the retirement expenses?