
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
\[4, \,\,\,\,\,\,5, \,\,\,\,\,\,8, \,\,\,\,\,\,12, \,\,\,\,\,\,13, \,\,\,\,\,\,16, \,\,\,\,\,\,20, \,\,\,\,\,\,24, \,\,\,\,\,\,25, \ldots,\]
where the first term is the smallest positive integer multiple of $4$, the next two terms are the next two smallest positive integers that are each one 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 $4$, and so on:
\[\underbrace{4}_{1\text{ term}}, \,\,\,\,\,\, \underbrace{5, \,\,\,\,\,\, 8}_{2\text{ terms}}, \,\,\,\,\,\, \underbrace{12, \,\,\,\,\,\, 16, \,\,\,\,\,\, 20}_{3\text{ terms}}, \ldots.\]
Determine $v_{2023}$.

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