Stable Matching Malicious Intent $(5$ Points$)$

So far, we have assumed that when people give their preference list as an input to the Gale$-$Shapley algorithm, they are acting truthfully. In other words, their preference list truly reflects their desire. However, what will happen if someone lied about their list? Could they achieve better results by manipulating it$?$

More concretely, consider a student S who prefers a university U to U$,$ yet both of them are low on the list. Could it be the case that by switching the order of U and U$,$ that S will match with U$$ who is preferred to both U and U$?$ Resolve this by proving that such an U and U$$ does not exist and there is no way for S to match with U$,$ or give a counterexample where S will match with U$.$