Mike wants to donate $$5,000,000$ to establish a fund to provide an annual scholarship in perpetuity. The fund will earn an interest rate of $5\mathrm{.}14\%$ p$.$a$.$ compounded half$-$yearly $($j$2=5\mathrm{.}14\%$ p$.$a$.)$ and the first scholarship will be first awarded $3\mathrm{.}5$ years after the date of the donation.

$($a$)$ What is the amount of the annual scholarship $($rounded to two decimal places$)?$

$($b$)$ Assume that two years after the donation, Mike needs to withdraw $$1,000,000$ from the fund and use the remaining amount to provide an annual scholarship in perpetuity. The time of the first scholarship will be unchanged $(3\mathrm{.}5$ years after the date of the donation$).$ What is the new annual scholarship amount $($rounded to two decimal places$)?$