
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
Petya has thought of two polynomials \( f(x) \) and \( g(x) \), each of the form \( a x^2 + bx + c \) (i.e., the degree of each polynomial does not exceed 2). During each turn, Vasya names a number \( t \), and Petya provides him with one of the values \( f(t) \) or \( g(t) \) (without specifying which one he provided). After \( n \) turns, Vasya must determine one of Petya's polynomials. For what minimum \( n \) does Vasya have a strategy that guarantees this outcome?

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