
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 $f(x)=\frac{ax^2}{\ln x}$, its tangent line at $x=e$ passes through the point $(1,e)$. (Here, $e=2.71828\ldots$)
(I) Find the maximum and minimum values of the function $f(x)$ on the interval $[e^{1/4}, e]$;
(II) If the equation $g(x)=tf(x)-x$ has two roots in the interval $[\frac{1}{e}, 1)\cup(1, e^2]$, find the range of the real number $t$.

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