
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
Let \(v_n\) be the \(n^\text{th}\) term of the sequence
\[2,\,\,\,\,\,\,4,\,\,\,\,\,\,7,\,\,\,\,\,\,8,\,\,\,\,\,\,11,\,\,\,\,\,\,14,\,\,\,\,\,\,15,\,\,\,\,\,\,18,\,\,\,\,\,\,21,\,\,\,\,\,\,24,\,\,\,\,\,\,25,\ldots,\]
where the first term is the smallest positive integer that is $2$ more than a multiple of $3$, the next two terms are the next two smallest positive integers that are each three more than a multiple of $3$, the next three terms are the next three smallest positive integers that are each four more than a multiple of $3$, the next four terms are the next four smallest positive integers that are each five more than a multiple of $3$, and so on:
\[\underbrace{2}_{1\text{ term}},\,\,\,\,\,\,\underbrace{4,\,\,\,\,\,\,7}_{2\text{ terms}},\,\,\,\,\,\,\underbrace{8,\,\,\,\,\,\,11,\,\,\,\,\,\,14}_{3\text{ terms}},\,\,\,\,\,\,\underbrace{15,\,\,\,\,\,\,18,\,\,\,\,\,\,21,\,\,\,\,\,\,24}_{4\text{ terms}},\,\,\,\,\,\,\underbrace{25,\ldots}_{5\text{ terms}},\,\,\,\,\,\,\ldots.\]
Determine \(v_{2008}\).

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