
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
24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \( a_k \) based on their ranking \( k \) in that round, where \( a_{k} \in \mathbb{N}_+, k = 1, 2, \ldots, n, a_1 > a_2 > \cdots > a_n \). After all the rounds are completed, the overall ranking is determined based on the total score each person has accumulated over the 12 rounds. Find the smallest positive integer \( n \) such that no matter the ranking in the penultimate round, at least 2 participants have the potential to win the championship.

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