
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
Find the function $f(x) = a\ln x + x^2$ (where $a$ is a real constant):
1. When $a = -4$, find the maximum value of $f(x)$ on the interval $[1, e]$ and the corresponding $x$ value.
2. When $x \in [1, e]$, discuss the number of roots of the equation $f(x) = 0$.
3. If $a > 0$, and for any $x_1, x_2 \in [1, e]$, it holds that $\left| f(x_1) - f(x_2) \right| \leq \left| \frac{1}{x_1} - \frac{1}{x_2} \right|$, find the range of the real number $a$.

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