John and Daphne are saving for their daughter Ellen's college education. Ellen just turned $10($at t $=0),$ and she will be entering college $8$ years from now $($at t $=8).$ College tuition and expenses at State U$.$ are currently $$14,500$ a year, but they are expected to increase at a rate of $3\mathrm{.}5\%$ a year. Ellen 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, John and Daphne have accumulated $$12,000$ in their college savings account $($at t $=0).$ Their long$-$run financial plan is to add an additional $$5,000$ 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 $9\%.$ How large must the annual payments at t $=5,6,$ and $7$ be to cover Ellen's anticipated college costs?

a$.$ $$3,415\mathrm{.}79$

b$.$ $$4,085\mathrm{.}71$

c$.$ $$5,868\mathrm{.}68$

d$.$ $$4,453\mathrm{.}43$

e$.$ $$5,384\mathrm{.}11$