$In$ january $2021,$ John borrowed $$29,000$ from a bank for his further studies. THe outstanding loan balance incurs a compound interest $of4\mathrm{.}5\%$ per year. The interest charged $is$ calculated and the outstanding balance $is$ updated $on$ the last day $of$ each year.

John plans $to$ pay back a fixed amount $of$ $$xon$ the first day $of$ each month, starting from the first month after his graduation. John will graduate from his studies $at$ the end $of2024$ and will make his first payment $on1st$ Januaury $2025.$

Show that the amount owed $on$ the last day $of2025$ after the interest $is$ added $is{1\mathrm{.}045}^5(29000)-1\mathrm{.}045(12x),$ and find the minimum amount, correct $to$ the nearest dollar, that John must pay each month, $ifhe$ wishes $to$ complete paying the loan $by$ the end $of$ December $2032.$