
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
Given that [ $r$ ] denotes the largest integer not greater than $r$, for any positive real number $x$, the set $A(x)$ is defined as: $A(x)=\left\{[n x] \mid n \in \mathbf{N}_{+}\right\}$.

Find all irrational numbers $\alpha$ greater than 1 such that if a positive real number $\beta$ satisfies $A(\alpha) \supset A(\beta)$, then $\frac{\beta}{\alpha}$ is an integer.

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