
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 any positive integer  $ d$ , in any  $ u_n$  consecutive odd positive integers, the number of them that can be divided by  $ d$  is not smaller than the number of odd integers among  $ 1, 3, 5, \ldots, 2n \minus{} 1$  that can be divided by  $ d$ .

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