
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
\[ 1, \,\,\,\,\, 5, \,\,\,\,\, 6, \,\,\,\,\, 10, \,\,\,\,\, 11, \,\,\,\,\, 15, \,\,\,\,\, 16, \,\,\,\,\, 20, \,\,\,\,\, 21, \,\,\,\,\, 25, \,\,\,\,\, 30, \ldots, \]
where the first term is the smallest positive integer that is \( 1 \) more than a multiple of \( 4 \), the next two terms are the next two smallest positive integers that are each \( 2 \) more than a multiple of \( 4 \), the next three terms are the next three smallest positive integers that are each \( 3 \) more than a multiple of \( 4 \), the next four terms are the next four smallest positive integers that are each \( 4 \) more than a multiple of \( 4 \), and so on:
\[\underbrace{1}_{1\text{ term}},\,\,\,\,\,\,\underbrace{5,\,\,\,\,\,\,10}_{2\text{ terms}},\,\,\,\,\,\,\underbrace{6,\,\,\,\,\,\,11,\,\,\,\,\,\,16}_{3\text{ terms}},\,\,\,\,\,\,\underbrace{20,\,\,\,\,\,\,25,\,\,\,\,\,\,30,\,\,\,\,\,\,35}_{4\text{ terms}},\,\,\,\,\,\,\underbrace{21, \ldots}_{5\text{ terms}},\,\,\,\,\,\,\ldots.\]
Determine \( v_{2008} \).

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