
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 $f(x) = a \ln x + x^2$ (where $a$ is a real constant).

(1) When $a = -4$, find the maximum value of the function $f(x)$ on the interval $[1, e]$ and the corresponding $x$ value;

(2) If $a > 0$, and for any $x_1$, $x_2 \in [\frac{1}{e}, \frac{1}{2}]$, we have $|f(x_1) - f(x_2)| \leq |\frac{1}{x_1} - \frac{1}{x_2}|$, find the range of values for the real number $a$.

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