
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 a positive integer \( n \), define the set \( A_{n} = \left\{ a \mid a \in \mathbb{Z}^{+}, a \leq n, \, n \mid (a^{n} + 1) \right\} \).
1. Find all positive integers \( n \) such that \( A_{n} \) is non-empty.
2. Find all positive integers \( n \) such that \( A_{n} \) is non-empty and \( |A_{n}| \) is even.

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