$($Comprehensive problem$)$ You would like to have $$73,000$ in $12$ years. To accumulate this amount, you plan $t$ deposit an equal sum in the bank each year that will earn $7$ 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 the lump$-$sum deposit be$?($Assume you can earn $7$ percent on this deposit.$)$

c$.$ At the end of year $5,$ you will receive $$15,000$ and deposit it in the bank in an effort to reach your goal of $$73,0$ at the end of year $12.$ In addition to the lump$-$sum deposit, how much must you invest in $12$ equal annual deposit reach your goal? $($Again$,$ assume you can earn $7$ percent on this deposit.$)$

b$.$ If you decide to make a large lump$-$sum deposit today instead of the annual deposits, how large should the lump$-$sum deposit be$?$

$$32412\mathrm{.}87($Round to the nearest cent.$)$

c$.$ If you deposit $$15,000$ received at the end of year $5$ in the bank, how much will it grow to in the account at the end of year $12?$

$

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