
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
Each of the numbers  $1$  up to and including  $2014$  has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number  $k$  of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of  $k$  that can be obtained.

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