
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 a positive odd integer.

1) Determine, as a function of \( n \), the largest integer \( r \) with the following property: for any real numbers \( a_{1}, \ldots, a_{n} \), there exist \( r \) integers \( 1 \leq i_{1} < i_{2} < \ldots < i_{r} \leq n \) such that for any integers \( k, \ell \) satisfying \( 1 \leq k, \ell \leq r \), we have \( a_{i_{k}} - a_{i_{\ell}} \neq 1 \).

2) Let \( \alpha \) be an irrational number. Determine, as a function of \( n \), the largest integer \( r \) with the following property: for any real numbers \( a_{1}, \ldots, a_{n} \), there exist \( r \) integers \( 1 \leq i_{1} < i_{2} < \ldots < i_{r} \leq n \) such that for any integers \( k, \ell \) satisfying \( 1 \leq k, \ell \leq r \), we have \( a_{i_k} - a_{i_{\ell}} \neq 1 \) and \( a_{i_k} - a_{i_{\ell}} \neq \alpha \).

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