James and Kennedy are saving for their daughter Miray's college education. Miray just turned $),$ and she will be entering college $8$ years from now $($at $t=8).$ College tuition and expenses at State U$.$ are currently $$13,000$ a year, but they are expected to increase at a rate of $3\mathrm{.}5\%$ a year. Miray should graduate in $4$ years$--$if she takes longer or wants to go to graduate school, she will be on her own. Tuition and other costs will be due at the beginning of each school year $($at $t=8,9,10,$ and $11).$

So far, James and Kennedy have accumulated $$10,000$ in their college savings account $($at $t=0).$ Their long$-$run financial plan is to add an additional $$5,500$ in each of the next $4$ years $($at $t=1,2,3,$ and $4).$ Then they plan to make $3$ equal annual contributions in each of the following years, $t=5,6,$ and $7.$ They expect their investment account to earn $7\%.$ How large must the annual payments at $t=5,6,$ and $7$ be to cover Miray's anticipated college costs?

a$.$ $$5,976\mathrm{.}56$

b$.$ $$4,650\mathrm{.}06$

c$.$ $$4,009\mathrm{.}24$

d$.$ $$5,585\mathrm{.}57$

e$.$ $$4,975\mathrm{.}56$