
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 $n$ be a given positive integer. Find the smallest positive integer $u_{n}$ such that for every positive integer $d$, any $u_{n}$ consecutive odd positive integers contain at least as many numbers divisible by $d$ as there are among $1, 3, 5, \cdots,$ and $2n-1$.

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