Prof. Business has a self$-$managed retirement plan through her University and would like to retire in $14$ years and wonders if her current and future planned savings will provide adequate future retirement income. Here are her information and goals.$$Prof$.$ Business wants a $23-$year retirement annuity that begins $14$ years from today with an equal annual payment equal to $$92,000$ today inflated at $3\mathrm{.}5\%$ annually over $14$ years. Her first retirement annuity payment would occur $14$ years from today. She realizes her purchasing power will decrease over time during retirement.$$Prof$.$ Business currently has $$450,000$ in her University retirement account. She expects these savings and any future deposits into her University and any other retirement account will earn $7\mathrm{.}5\%$ compounded annually. Also, she expects to earn this same $7\mathrm{.}5\%$ annual return after she retires.$$

Answer the following questions to help Prof. Business finalize her retirement planning.

$1)$ What is Prof. Business desired annual retirement income?

$2)$How much will Prof. Business need $14$ years from today to fund her desired retirement annuity?

$3)$In addition to the $$450,000$ balance today, Prof. Business will fund her future retirement goal from question $2$ by making $14$ annual equal deposits at $7\mathrm{.}5\%$ compounded annually into her retirement accounts starting a year from today $($the last deposit will be made when Prof. Business retires$).$ How large does this annual deposit need to be in addition to the initial $$450,000$ invested in Prof. Business retirement fund?

$4)$ This annual figure from question $3$ makes Prof. Business feel a little anxious about her future planned retirement since her current annual contribution is $$14,600.$ Also, Prof. Business annual retirement account contribution is based on a percentage of her salary and will increase as her salary increases. However, Prof. Business is worried about her purchasing power eroding during retirement. She would like her first retirement withdrawal to be equal to the amount you found in question $1,$ and then she increases each successive retirement withdrawal by $3\mathrm{.}5\%$ annually over the remaining $22$ withdrawals. How much will Prof. Business need now at retirement given Prof. Business $7\mathrm{.}5\%$ expected return?

Explain each part with steps and details.