
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
Let  $n$  be a given positive integer. Alice and Bob play a game. In the beginning, Alice determines an integer polynomial  $P(x)$  with degree no more than  $n$ . Bob doesn’t know  $P(x)$ , and his goal is to determine whether there exists an integer  $k$  such that no integer roots of  $P(x) = k$  exist. 

In each round, Bob can choose a constant  $c$ . Alice will tell Bob an integer  $k$ , representing the number of integer  $t$  such that  $P(t) = c$ . Bob needs to pay one dollar for each round. Find the minimum cost such that Bob can guarantee to reach his goal.

*Proposed by ltf0501*

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