Lee plans to retire in $20$ years with a nest egg of $$6$M$.$ He has already saved up $$400,000$ in an investment account that generates an annual rate of return of $12\%,$ compounded monthly.

$($a$)$ Numerically show whether Lee$$s investment account balance will reach $$6$M in $20$ years, based on the information provided above. $<...>$

$($b$)$ The correct answer for part $($a$)$ indicates that Lee$$s investment account will fall short of his retirement goal of $$6$M in $20$ years. Thus, he continues his pursuit by making additional fixed contributions at the beginning of each quarter to the same investment account until he retires $20$ years later. Calculate the amount of his quarterly contribution needed for achieving his goal. $<$ $$4,890>$

$($c$)$ Lee now retires with $$6$M that is transferred to a retirement account offering an annual return of $6\mathrm{.}6\%,$ compounded monthly. He withdraws monthly from the retirement account to cover his living expenses that grow at a monthly inflation rate of $0\mathrm{.}3\%.$ Lee takes out his first withdrawal of $$22,500$ at the end of the month. Calculate the amount of Lee$$s inheritance upon his death after enjoying $25$ years of retirement. $<...>$