$($Complex present value$)$ You would like to have $$$$33,000$ in $13$ years. To accumulate this$$ amount, you plan to deposit each year an equal sum in the$$ bank, which will earn $8$ percent interest compounded annually. Your first payment will be made at the end of the year.

a$.$ How much must you deposit annually to accumulate this$$ amount?

b$.$ If you decide to make a large$$ lump$-$sum deposit today instead of the annual$$ deposits, how large should this$$ lump$-$sum deposit$$ be$?($Assume you can earn $8$ percent on this$$ deposit.$)$

c$.$ At the end of $6$ years you will receive $$$$8,000$ and deposit this in the bank toward your goal of $$$$33,000$ at the end of $13$ years. In addition to this$$ deposit, how much must you deposit in equal annual deposits to reach your$$ goal? $($Again assume you can earn $8$ percent on this$$ deposit.