Alice and Bob play the following game with \( S=\{1,\dots, 777\} \).

Alice picks a number \(x \in S\) without telling anyone and Bob will guess what the number is at the end of the game. Alice is malicious so that she can always change her number \(x\) at any time until the end of the game.

In each round, Bob picks a subset \(T\subseteq S\) and asks a following question to Alice: “is your \(x\) belong to \(T\)?” Alice must say either Yes or No. At the end of the game, Bob guesses her \(x\) first and then Alice reveals her number \(x\) (Alice can still change her number after she listen to Bob’s guess and before revealing her number). According to her final number \(x\), each of her previous answers are determined to be either a truth or a lie.

Bob wins if Alice end up lying more than three times or his answer is correct. Alice wins if Bob’s answer is wrong and at most three of her answers are lies. Prove that if a game consists of twenty rounds, then no matter what Bob does Alice can always win.

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