
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
A non-empty set  $ S$  of positive integers is said to be *good* if there is a coloring with  $ 2008$  colors of all positive integers so that no number in  $ S$  is the sum of two different positive integers (not necessarily in  $ S$ ) of the same color. Find the largest value  $ t$  can take so that the set  $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$  is good, for any positive integer  $ a$ .

<details><summary>P.S.</summary>I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...</details>

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