
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
Let the function \( f(x)=\frac{x}{\ln x}-ax \).
1. Find the minimum value of the real number \( a \) such that \( f(x) \) is a monotonically decreasing function on the interval \( (1,+\infty) \).
2. Determine the range of \( a \) such that there exist \( x_1, x_2 \in [e, e^2] \) satisfying
   \[
   f(x_1) \leq f'(x2) + a.
   \]

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