You have just inherited a large sum of money and you are trying to determine how much you should save for retirement and how much you can spend now. For retirement, you will deposit today $($January $1$st$,2024)$ a lump sum in a bank account paying $10\%$ compounded annually. You dont plan on touching this deposit until you retire in five years $($December $31$st$,2028),$ and you plan on living for $20$ additional years and then drop dead on December $31$st$,2048.$ During your retirement, you would like to receive income of $$60,000$ per year to be received the first day of each year, with the first payment on January $1$st$,2029,$ and the last payment on January $1$st$,2048.$ Complicating this objective is your desire to have one final three$-$year fling, during which time youd like to track down all the original members of The Harry Potter Series and get their autographs. To finance this, you want to receive $$300,000$ on January $1$st$,2045,$ and nothing on January $1$st$,2046,$ and January $1$st$,2047,$ as you will be on the road. In addition, after you pass on $($January $1$st$,2049),$ you would like to have a total of $$100,000$ to leave to your children.

Required:

How much must you deposit in the bank at $10\%$ p$.$a$.$ compounded annually on January $1$st$,2024,$ in order to achieve your goal. Use a timeline in order to answer this question.

$(30$ marks$)$