
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 a function $f(x) = \frac{a}{x^{2}} + \ln x, a \in \mathbb{R}$.
(1) Discuss the monotonicity of the function $f(x)$;
(2) If for any $x \in \left[ \frac{1}{2}, 2 \right]$, the inequality $f(x) \geq \frac{1}{x}$ always holds, find the range of the real number $a$.

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