A couple plans to purchase a vacation home. The bank requires a $10\%$ down payment on the $$400,000$ vacation home. The couple will finance the rest of the cost with a fixed$-$rate mortgage at $6\%$ annual interest with monthly payments over $30$ years.Complete the parts below. Do not round any intermediate computations. Round your final answers to the nearest cent if necessary.$($a$)$ Find the required down payment.Is$($b$)$ Find the amount of the mortgage.$($c$)$ Find the monthly payment.