
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
Discriminant the cat proposes the following game:

- You choose a finite sequence (of any length and secret) of natural numbers $\left(a_{0}, \ldots, a_{n}\right) \in \mathbb{N}^{n+1}$.
- In each round of the game, Discriminant gives you a natural number $\beta \in \mathbb{N}$ and you respond with the value of $\sum_{k=0}^{n} a_{k} \beta^{k}$.
- The game continues as long as Discriminant has not guessed the sequence of numbers and fewer than $N$ rounds have taken place.
- If at the end of the game Discriminant has guessed the sequence of numbers, he wins; otherwise, he loses.

For what values of $N$ would you accept to play this game?

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