A state lottery gives a winner the choice of receiving the winning amount in equal monthly payments for $20$ years or receiving a lump sum equal to the present value of an annuity with future value equal to the winnings. The winner selecting monthly payments will receive $$6,000,000/240=$ $$25,000\mathrm{.}00$ each month for each million dollars of winnings. $($Round your final answers to two decimal places.$)$

$($a$)$ Find the present value of an annuity with monthly payments of $$25,000\mathrm{.}00,$ at an interest rate of $4\mathrm{.}1\%$ for $20$ years, for the winner who wants a lump$-$sum payment.

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$($b$)$ In order for the lottery to be more profitable, it is decided to pay the winnings in equal monthly payments for $25$ years. Find the monthly payments of $$6$ million in winnings.

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Find the present value of an annuity with those monthly payments at $4\mathrm{.}1\%$ for $25$ years.

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