
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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?

Present the answer in LaTex format: \boxed{Your answer}
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