
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
Given a set \( S = \{1, 2, 3, \cdots, N\} \) and four 500-element subsets \( A_{1}, A_{2}, A_{3}, A_{4} \) such that for any \( x, y \in S \), there exists an \( i \in \{1, 2, 3, 4\} \) where \( x, y \in A_{i} \). Find the maximum possible value of the positive integer \( N \).

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