
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
\[1,\,\,\,\,\,\,3,\,\,\,\,\,\,7,\,\,\,\,\,\,9,\,\,\,\,\,\,13,\,\,\,\,\,\,17,\,\,\,\,\,\,19,\,\,\,\,\,\,23,\,\,\,\,\,\,27,\,\,\,\,\,\,31,\,\,\,\,\,\,33,\ldots,\]
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 $3$ 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 $3$ more than a multiple of $4$, and so on:
\[\underbrace{1}_{1\text{ term}},\,\,\,\,\,\,\underbrace{3,\,\,\,\,\,\,7}_{2\text{ terms}},\,\,\,\,\,\,\underbrace{9,\,\,\,\,\,\,13,\,\,\,\,\,\,17}_{3\text{ terms}},\,\,\,\,\,\,\underbrace{19,\,\,\,\,\,\,23,\,\,\,\,\,\,27,\,\,\,\,\,\,31}_{4\text{ terms}},\,\,\,\,\,\,\underbrace{33,\ldots}_{5\text{ terms}},\,\,\,\,\,\,\ldots.\]
Determine \( v_{1004} \).

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