Jack is planning on retiring when he turns $65,$ and today he celebrated his $45$th birthday. Starting one year after retirement, Jack will receive $20$ annual payments from the retirement fund. The first of these payments will be $20,000,$ and each successive payment will be $5\%$ greater than the one prior. To make his retirement plans work out as planned, he accordingly makes $20$ annual deposits with the first occurring on his $46$th birthday and the last occurring on his $65$th birthday. The first of these annual deposits is $x,$ and each successive deposit is $500$ greater than the one prior. The annual effective interest rate for the retirement fund is $4\%$ for the first $20$ years and $5\%$ thereafter. The fund was opened today on his $45$ th birthday. Find $x.$