
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
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)?

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