
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
\[2,\,\,\,\,\,\,7,\,\,\,\,\,\,12,\,\,\,\,\,\,17,\,\,\,\,\,\,22,\,\,\,\,\,\,27,\,\,\,\,\,\,32,\,\,\,\,\,\,37,\,\,\,\,\,\,42,\,\,\,\,\,\,47,\,\,\,\,\,\,52,\ldots,\]
where the first term is $2$, the smallest positive integer that is $2$ more than a multiple of $5$. The next five terms are the next five smallest positive integers that are each $2$ more than a multiple of $5$, the next eight terms are the next eight smallest positive integers that are each $2$ more than a multiple of $5$, and so on:
\[\underbrace{2}_{1\text{ term}},\,\,\,\,\,\,\underbrace{7,\,\,\,\,\,\,12,\,\,\,\,\,\,17,\,\,\,\,\,\,22}_{5\text{ terms}},\,\,\,\,\,\,\underbrace{27,\,\,\,\,\,\,32,\,\,\,\,\,\,37,\,\,\,\,\,\,42,\,\,\,\,\,\,47,\,\,\,\,\,\,52,\,\ldots}_{8\text{ terms}},\,\,\,\,\,\,\ldots.\]
Determine $u_{2023}$.

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