
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
There are 12 stages in a racing tournament and $n$ participants. After each stage, all participants receive points $a_{k}$ based on their position $k$ ($a_{k}$ are natural numbers, and $a_{1}>a_{2}>\ldots>a_{n}$). For what is the smallest $n$ such that the tournament organizer can choose numbers $a_{1}, \ldots, a_{n}$ so that after the penultimate stage, at least two participants will still have a chance to take first place, regardless of the possible distribution of positions?

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