Question $3(25$ pts$)$

On her $25$th birthday, a young woman enginecr decides to start saving toward building up a retirement fund that pays $8\%$ interest compounded quartely $($the market interest rate$).$ She feels that $$1,000,000$ worth of purchasing power in today's dollars will be adequate for her at the age of $65.$ Assume a general inflation rate of $6\%$ per year.

$($a$)$ If she plans to save by making $160$ equal quarterly deposits, what should be the amount of her quarterly deposit in actual dollars? Assume the first deposit is made at the end of first quarter.

$($b$)$ If she plans to save by making end$-$of$-$the$-$year deposits, increasing by $$1,000$ over each subsequent year, how much would her first deposit be in actual dollars?