
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 \( a \) is a positive number, if the distinct positive real numbers \( x \) and \( y \) satisfy the equation \( \left(2x- \frac{y}{e}\right)·\ln \frac{y}{x}= \frac{x}{ae} \), determine the minimum value of \( a \).

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