
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,2009\}$ , choose  $ k$  pairs  $ \{a_i,b_i\}$  with  $ a_i<b_i$  so that no two pairs have a common element. Suppose that all the sums  $ a_i\plus{}b_i$  are distinct and less than or equal to  $ 2009$ . Find the maximum possible value of  $ k$ .

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