
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
Identify all such numbers $\alpha$ such that the numbers $[\alpha],[2 \alpha],[3 \alpha], \ldots,[N \alpha]$, where $N$ is a fixed natural number, are all distinct, and the numbers $\left[\frac{1}{\alpha}\right],\left[\frac{2}{\alpha}\right],\left[\frac{3}{\alpha}\right], \ldots,\left[\frac{N}{\alpha}\right]$ are also all distinct.

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