
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 the function $f(x) = x\ln{x}$ (where $e$ is an irrational number, $e \approx 2.718$),  
(1) Find the equation of the tangent line to the function $f(x)$ at the point $(e, f(e))$;  
(2) Let the real number $a > \frac{1}{2e}$, find the minimum value of the function $f(x)$ on the interval $[a, 2a]$.

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