
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
For any positive integer \(n\), let \(N_{n}\) be the set of integers from 1 to \(n\), i.e., \(N_{n}=\{1,2,3, \cdots, n\}\). Now assume that \(n \geq 10\). Determine the maximum value of \(n\) such that the following inequality
\[ \mathrm{max}_{\substack{a, b \in A \\ a \neq b}} |a-b| \leq 10 \]
holds for each \(A \subseteq N_{n}\) with \(|A| \geq 10\).

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