$[$Maximum mark: $7]$

$22$N$\mathrm{.}1.$SL$.$TZ$0\mathrm{.}7$

On $1$ December $2022,$ Laviola invests $800$ euros $($EUR$)$ into a savings account which pays a nominal annual interest rate of $7\mathrm{.}5\%$ compounded monthly. At the end of each month, Laviola deposits an additional EUR $500$ into the savings account.

At the end of $k$ months, Laviola will have saved enough money to withdraw EUR $10000.$

$($a$)$ Find the smallest possible value of $k,$ for $kin{Z}^{+}.$

$($b$)$ For this value of $k,$ find the interest earned in the savings account.

Express your answer correct to the nearest EUR.

$[3]$