
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 \( A_{1}, A_{2}, \cdots, A_{k} \) be \( k \) different subsets of the set \( \{1, 2, \cdots, 2016\} \) such that for any positive integers \( i \) and \( j \) where \( 1 \leq i < j \leq k \), the intersection \( A_{i} \cap A_{j} \) is always an arithmetic sequence. Find the maximum value of \( k \).

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