On her $28$th birthday, a young woman engineer decides to start saving toward building up a retirement fund that pays $9\%$ interest compounded monthly $($the market interest rate$).$ She feels that $$1,200,000$ worth of purchasing power in today$$s dollars will be adequate to see her through her sunset years after her $65$th birthday. Assume a general inflation rate of $6\%$ per year.

$($a$)$ Of she plan to save by making $444$ equal monthly deposits, what should be the amount of her monthly deposits in actual dollars? Assume the first deposit is made at the end of the first month.

$($b$)$ If she plan to save by making end$-$of$-$the$-$year$-$deposits, increasing by $$1,200$ over each subsequent year, how much would her first deposit be in actual dollars?