$(25$ points$)$ You begin an investment plan by putting $$10000$ in an account that you assume will earn $8\%$ annually. For the next $25$ years, you add $$\frac{5000}{y}.$ In anticipation of buying a house with a $15-$year mortgage, you expect to need a one$-$time down payment of $$55000$ at the end of year $8.$ You anticipate being able to make the monthly mortgage payment without affecting the yearly contribution to the savings plan. The discrete, nondiscounted cash flow diagram for this situation is shown below $($figures in $ thousand$).$

$($a$)$ Will you have the down payment at the end of year $8?$

$($b$)$ What will be the value of your savings account at the end of year $25?($Hint: you have to discount the values, but not to year $0.)$