Newman Weatherspoon, a super salesman contemplating retirement on his fifty$-$fifth birthday, decides to create a fund on an $9\%$ basis that will enable him to withdraw $$16,000$ per year on June $30,$ beginning in $2029$ and continuing through $2032.$ To develop this fund, Newman intends to make equal contributions on June $30$ of each of the years $2025-2028.$

$($a$)$

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How much must the balance of the fund equal on June $30,2028,$ in order for Newman to satisfy his objective? $($Round factor values to $5$ decimal places, e$.$g$.1\mathrm{.}25124$ and final answer to $0$ decimal places, e$.$g$.458,581.)$

Balance of the fund equal on June $30,2028$ $

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$($b$)$

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