$($Comprehensive problem$)$ You would like to have $$70,000$ in $13$ years. To accumulate this amount, you plan to 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 $$20,000$ and deposit it in the bank in an effort to reach your goal of $$70,000$ 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 $7$ percent on this deposit$)$

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

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

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

$($Round to the nearest cent$)$

c$.$ If you deposit $$20,000$ received at the end of year $5$ in the bank, how much will it grow to in the account at the end of year $13?$

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

In addition to the lump$-$sum deposit, how much must you invest in $13$ equal annual deposits to reach your goal?

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