$($Solving a comprehensive problem$)$ You would like to have $$80,000$ in $13$ years. To accumulate this amount, you plan to deposit an equal sum in the bank each year that will earn $10$ 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 $10$ 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 $$80,000$ at the end of year $13.$ In addition to the lump$-$sum deposit, how much must you invest in $13$ equal annual deposits to reach your goal? $($Again$,$ assume you can earn $10$ percent on this deposit.$)$

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

$

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