
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 $f(x) = \ln x$, and $F(x) = x + \frac{1}{x} + af'(x)$:

(I) When $a=1$, find the extreme value of $M(x) = F(x) - f(x)$.

(II) When $a=0$, for any $x > 0$, the inequality $\frac{1}{F(x)} \leq \frac{1}{2 + m[f(x)]^2}$ always holds; find the range of real numbers $m$.

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