
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 odd positive 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 $1 \leqslant i_{1}<i_{2}<\ldots<i_{r} \leqslant n$ such that for all integers $k, \ell$ satisfying $1 \leqslant k, \ell \leqslant r$, 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 $1 \leqslant i_{1}<i_{2}<\ldots<i_{r} \leqslant n$ such that for all integers $k, \ell$ satisfying $1 \leqslant k, \ell \leqslant r$, 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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