
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 $f(x) = \frac{1 + 2\ln x}{x^2}$.

(1) Find the maximum value of $f(x)$;

(2) Let $g(x) = ax^2 - 2\ln x$, when $x > 0$, the maximum value of $f(x)$ is $M$, $g(x) = M$ has two different roots, find the range of $a$;

(3) There exist $x_1, x_2 \in (1, +\infty)$, and $x_1 \neq x_2$, such that $|f(x_1) - f(x_2)| \geq k|\ln x_1 - \ln x_2|$ holds, find the range of $k$.

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