
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 $x \in \mathbb{R}$, define: $A(x)$ represents the smallest integer not less than $x$. For example, $A(\sqrt{3})=2, A(-0.4)=0$, and $A(-1.1)=-1$.  
(Science) If $A(2x \cdot A(x))=5$, then the range of values for the positive real number $x$ is \_\_\_\_\_\_.

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