
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 $u_n$ be the $n^\text{th}$ term of the sequence
\[2,\,\,3,\,\,6,\,\,7,\,\,10,\,\,13,\,\,14,\,\,17,\,\,20,\,\,23,\,\,24,\ldots,\]
where the first term is the smallest positive integer that is $2$ more than a multiple of $4$, the next two terms are the next two smallest positive integers that are each three more than a multiple of $4$, the next three terms are the next three smallest positive integers that are each four more than a multiple of $4$, the next four terms are the next four smallest positive integers that are each five more than a multiple of $4$, and so on:
\[\underbrace{2}_{1\text{ term}},\,\,\,\,\,\,\underbrace{3,\,\,\,\,\,\,6}_{2\text{ terms}},\,\,\,\,\,\,\underbrace{7,\,\,\,\,\,\,10,\,\,\,\,\,\,13}_{3\text{ terms}},\,\,\,\,\,\,\underbrace{14,\,\,\,\,\,\,17,\,\,\,\,\,\,20,\,\,\,\,\,\,23}_{4\text{ terms}},\,\,\,\,\,\,\underbrace{24,\ldots}_{5\text{ terms}},\,\,\,\,\,\,\ldots.\]
Determine $u_{2008}$.

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