
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
\[ 1, \,\, 3, \,\, 7, \,\, 8, \,\, 12, \,\, 13, \,\, 19, \,\, 20, \,\, 24, \,\, 25, \,\, 31, \,\, 32, \,\, 38, \,\, 39, \ldots \]
where the first term is the smallest positive integer that is one more than a multiple of 4, the next two terms are the next two smallest positive integers that are each two more than a multiple of 4, the next three terms are the next three smallest positive integers that are each three more than a multiple of 4, the next four terms are the next four smallest positive integers that are each four more than a multiple of 4, and so on:
\[ \underbrace{1}_{1\text{ term}}, \,\, \underbrace{3, \,\, 7}_{2\text{ terms}}, \,\, \underbrace{8, \,\, 12, \,\, 13}_{3\text{ terms}}, \,\, \underbrace{19, \,\, 20, \,\, 24, \,\, 25}_{4\text{ terms}}, \,\, \underbrace{31, \ldots}_{5\text{ terms}}, \,\, \ldots. \]
Determine \( v_{2010} \).

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