
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
Petia has devised two polynomials \( f(x) \) and \( g(x) \), each of the form \( a x^{2} + b x + c \) (i.e., each polynomial does not exceed degree 2). In his turn, Vasya names a number \( t \), and Petia informs him of one of the values \( f(t) \) or \( g(t) \) (without specifying which one he is reporting). After \( n \) turns, Vasya must determine one of Petia's polynomials. What is the smallest \( n \) for which Vasya has a strategy that guarantees he can achieve this?

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