Congratulations! Today is your $21$st birthday. You just started working full$-$time, earning $$100,000$ per year. Your goal is to have $$5$ million in your $401($k$)$ plan by your $66$th birthday $($i$.$e$.,45$ years from today$).$ Assume $3\%$ inflation per year, and that you start with nothing. If you can earn $10\%$ per year annualized in an S&P $500$ mutual fund, after all expenses, inside a $401($k$)$ with a dollar for dollar match up to $10\%$ of your income, how much would you need to save each month to have that $$5$ million:

a$)$ If that $$5$ million is in future $($i$.$e$.,$ nominal$)$ dollars $($ignore taxes$)$;

b$)$ If that $$5$ million is in today$$s equivalent purchasing power future dollars $($ignore taxes$)$;

c$)$ If your marginal tax rate is $28\%$ federal plus $7\%$ state, what would the after$-$tax cost of your investments be for $($i$)$ and $($ii$)$ if you relied on the employer match for $(1)/(2)$ of your monthly contributions?

d$)$ In retirement, you plan to draw $$10,000$ per month of principal from your investments. If you were $100\%$ invested in stock mutual funds $($like the S&P $500),$ would you get more or less than the rate of return on the S&P $500-$ and why?

$($Please show all work$)$