Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him the value of $a_{1}x_{1}+\cdots+a_{n}x_{n}$. Then Bob chooses another list of positive integers $b_{1}$, $\cdots$, $b_{n}$ and asks Alice for $b_{1}x_{1}+\cdots+b_{n}x_{n}$. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?