
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 $f(x)= \frac{bx}{\ln x}-ax$.
(1) If $a=0$, find the interval where $f(x)$ is monotonically increasing.
(2) When $b=1$, if there exist $x_{1}$, $x_{2}∈[e,e^{2}]$ such that $f(x_{1})≤f′(x_{2})+a$ holds, find the minimum value of the real number $a$. (Note: $e$ is the base of the natural logarithm)

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