Congrats! You just won the Engineering Economy lottery! Your award will be: A starting sum of $$80,000$ at the end of year $1,$ with yearly increases of $4\%$ each year! Given your financial expertise, you plan on opening a retirement account, where you will deposit $15\%$ of your winnings each year for the next $30$ years. Keep in mind, your retirement account has an annual interest rate of $8\mathrm{.}20\%.$

a$)$ How much money will be in your retirement account at the end of $30$ years? $($Hint: this is a geometric gradient problem$)$

b$)$ How much can you withdraw from that account each year in retirement for $15$ years? Assume you will withdraw the same amount each year. $($Hint: this is a uniform annuity problem$)$