
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 of integers \( \{1, 2, 3, \ldots, 3009\} \), choose \( k \) pairs \( \{a_i, b_i\} \) with \( a_i < b_i \) such that no two pairs have a common element. Assume that all the sums \( a_i + b_i \) are distinct and less than or equal to 3009. Determine the maximum possible value of \( k \).

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