
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 \( n \in \mathbf{N}^{*} \). Consider the set \( S = \{1, 2, \ldots, 2n\} \) and its \( k \) subsets \( A_{1}, A_{2}, \ldots, A_{k} \) that satisfy the following conditions:

1. For any \( i \neq j \) (with \( i, j \in \{1, 2, \ldots, k\} \)), the intersection \( A_{i} \cap A_{j} \) has exactly an odd number of elements.
2. For any \( i \) (where \( i = 1, 2, \ldots, k \)), \( i \notin A_{i} \).
3. If \( i \in A_{j} \), then \( j \in A_{i} \).

Determine the maximum value of \( k \).

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