
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
\[ 1, \,\,\,\,\,\, 4, \,\,\,\,\,\, 8, \,\,\,\,\,\, 11, \,\,\,\,\,\, 14, \,\,\,\,\,\, 18, \,\,\,\,\,\, 21, \,\,\,\,\,\, 24, \,\,\,\,\,\, 28, \,\,\,\,\,\, 31, \,\,\,\,\,\, 34, \,\,\,\,\,\, 38, \,\,\,\,\,\, 41, \ldots, \]
where the first term is the smallest positive integer that is \( 1 \) more than a multiple of \( 3 \), the next two terms are each \( 2 \) more than a multiple of \( 3 \), the next three terms are each \( 3 \) more than a multiple of \( 3 \), continuing in this manner, increasing the adding difference as:
\[ \underbrace{1}_{1\text{ term}}, \,\,\,\,\,\, \underbrace{4, \,\,\,\,\,\, 8}_{2\text{ terms}}, \,\,\,\,\,\, \underbrace{11, \,\,\,\,\,\, 14, \,\,\,\,\,\, 18}_{3\text{ terms}}, \,\,\,\,\,\, \ldots \]
Determine \( v_{2008} \).

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