
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 $f(x) = a\ln x - bx^2$.
(I) Given $a=2$ and $b= \frac{1}{2}$, find the maximum value of $f(x)$ in the interval $\left[\frac{1}{e}, e\right]$.
(II) When $b=0$, if the inequality $f(x) \geq m + x$ holds for all $a \in \left[0, \frac{3}{2}\right]$ and $x \in (1,e^2]$, determine the range of the real number $m$.

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