
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
Given \( N \in \mathbb{N} \):

Discriminant the cat proposes the following game:

- You choose a finite sequence (of any length and secret) of integers \((a_{0}, \ldots, a_{n}) \in \mathbb{N}^{n+1}\).
- In each round of the game, Discriminant gives you a real 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 integers and less than \(N\) rounds have taken place.
- If by the end of the game Discriminant has guessed the sequence of integers, he wins; otherwise, he loses.

For which values of \(N\) would you accept playing this game?

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