
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
For real positive numbers  $x,$  the set  $A(x)$  is defined by 
\[A(x)=\{[nx]\ | \ n\in{ \mathbb{N}}\},\] where  $[r]$  denotes the greatest integer not exceeding real numbers  $r.$ Find all irrational numbers  $\alpha >1$  satisfying the following condition.
Condition: If  positive real number  $\beta$  satisfies  $A(\alpha)\supset A(\beta),$  then  $\frac{\beta}{\alpha}$  is integer.

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