The owner of an antique shop, Sarah, displayed a rare Victorian$-$era writing desk in the storefront window with a price tag of $$800.$ Mark, a regular customer, saw the desk on Monday and was interested in purchasing it$.$ Mark entered the shop and said to Sarah, $$I$\text{'}$ll buy the writing desk for $$700.$ Sarah, considering Mark's offer, responded, $$I appreciate your interest, but I can only sell it for the marked price of $$800.$ Mark, determined to negotiate, asked, $$How about we meet in the middle at $$750?$ Sarah, unsure but wanting to accommodate a loyal customer, replied, $$Let me think about it$,$ and I'll get back to you$.$ The next day $($Tuesday$)$ morning, Mark received a text from Sarah, stating, $$I am seriously considering your offer. Please let me know if you will be able to pay cash for the purchase$.$ Mark got excited receiving the text, and replied, "Great! I will be able to pay cash for the deal. I'll come tomorrow to complete the purchase$.$ However, later that day $($Tuesday$),$ another customer, Brooke, was also interested in purchasing the writing desk. But Brooke wanted to consult with her husband before the purchase. Going back home, Brooke sent an email $($on Tuesday afternoon$)$ to Sarah saying that she would be happy to purchase the writing desk for $$800.$ Sarah immediately replied to the email saying, $$Wonderful$!$ The writing desk is yours. Please collect the desk anytime at your convenience$.$ However, the email was not sent to Brooke due to a technical problem with Sarah$$s web server. On Wednesday morning, Brooke changed his mind and sent another email to Sarah saying $$I am not interested in purchasing the writing desk anymore. You can sell the desk to any interested customer$.$ On Wednesday morning, Mark came to the shop with cash of $$750$ to pick up the writing desk. Sarah said, $$We don$$t have any deal. I had sold the desk to another customer already$.$ Mark, upset with the knockback, insists that they had a binding agreement at $$750$ and that Sarah cannot go back on their deal. On the other hand, Sarah believes that she already sold the writing desk to Brooke and, therefore, Brooke has no right to turn back. Examine whether a valid contract was formed at any point, considering the negotiations and the subsequent communications of Mark and Brooke with Sarah. Justify your response, citing case laws and statutes whenever relevant. When answering this question, fully consider the relevant principles, rules, and propositions in the Law of Contract.