
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 $f'(x)$ is the derivative of the function $f(x)$, and it satisfies $xf'(x)-xf(x)-e^{x}=0$ in the domain $(0,+\infty)$, and $f(1)=2e$. If $f(1-\frac{1}{2a})≤e^{\frac{1}{e}}$, then the range of real number $a$ is ____.

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