
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 $f(x) = a \ln x - bx^2$.

(I) If the function $f(x)$ is tangent to the line $y = -\frac{1}{2}$ at $x = 1$, find the maximum value of the function $f(x)$ on the interval $[\frac{1}{e}, e]$.

(II) When $b = 0$, if the inequality $f(x) \geq m + x$ holds for all $a \in [0, \frac{3}{2}]$ and $x \in (1, e^2]$, find the range of values for the real number $m$.

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