$($Complex present value$)$ You would like to have $$54,000$ in $9$ years. To accumulate this amount, you plan to deposit each year an equal sum in the bank, which will earn $9$ 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 $9$ percent on this deposit.$)$

c$.$ At the end of $4$ years you will receive $$11,000$ and deposit this in the bank toward your goal of $$54,000$ at the end of $9$ years. In addition to this deposit, how much must you deposit in equal annual deposits to reach your goal? $($Again assume you can earn $9$ percent on this deposit$)$

a$.$ How much must you deposit annually to accumulate $$54,000$ in $9$ years?

$ $($Round to the nearest cent.$)$