
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 and Vasya are playing a game. First, Petya thinks of a polynomial \( P(x) \) with integer coefficients. Then, several moves are made. On a move, Vasya pays Petya one ruble and names any integer \( a \) of his choice that he has not previously named, and Petya responds with the number of integer solutions to the equation \( P(x) = a \). Vasya wins as soon as Petya gives the same number twice (not necessarily consecutively). What is the minimum number of rubles Vasya needs to guarantee a win?

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