
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 an integer such that $1 \leq n \leq 30$. Ariane and Bérénice are playing the following game. First, Ariane chooses $n$ distinct numbers from the integers $1, 2, \ldots, 30$. Then, based on Ariane's choices, Bérénice chooses an integer $d \geq 2$. Bérénice wins if $d$ divides at least two of the numbers chosen by Ariane. Otherwise, Ariane wins.

For which values of $n$ can Ariane ensure she will win?

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