
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 are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial  $f(x)$  with the following properties:

(a) for any integer  $n$ ,  $f(n)$  is an integer;
(b) the degree of  $f(x)$  is less than  $187$ .

Alice knows that  $f(x)$  satisfies (a) and (b), but she does not know  $f(x)$ . In every turn, Alice picks a number  $k$  from the set  $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of  $f(k)$ . Find the smallest positive integer  $N$  so that Alice always knows for sure the parity of  $f(0)$  within  $N$  turns.

*Proposed by YaWNeeT*

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