During $2024,$ Christian and Danielle, a married couple, decided to sell their residence, which had a basis of $$200,000.$ They had owned and occupied the residence for $20$ years. To make it more attractive to prospective buyers, they had the inside painted in April at a cost of $$5,000$ and paid for the work immediately. They sold the house in May for $$800,000.$ Broker's commissions and other selling expenses amounted to $$50,000.$ The couple purchased a new residence in July for $$400,000.$ What is the recognized gain and the adjusted basis of the new residence?

a$.$$$45,000$ and $$400,000.$

b$.$$$550,000$ and $$800,000.$

c$.$$$50,000$ and $$400,000.$

d$.$$$100,000$ and $$600,000.$