As of late summer $2023,$ the average $30-$year fixed rate mortgage rate was $7\mathrm{.}18\%$ The median home sales price in the United States for the second quarter of $2023$ was around $$416,000.$

a$)$ Given a $20\%$ down payment, what is the monthly payment required to purchase the median home at the average $30-$year interest rate?

b$)2$ years ago, the median home sales price was nearly equivalent to today, but the average interest rate was $2\mathrm{.}85\%.$ What would have been the monthly payment on a $$416,000$ home purchased with a $20\%$ down payment in $2021?$

c$)$ At $2023$ interest rates, what is the purchase price of a home that yields the same monthly payment as the home in part b$?$ Assume you still have $$83,200$ as a down payment.