
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 \( f(x) = \ln x + \frac{a}{x} \) where \( a \in \mathbf{R} \):

(1) Determine the monotonicity of \( f(x) \) within its domain.

(2) If \( f(x) < 2x^2 \) always holds for \( x \in \left(\frac{1}{2}, +\infty \right) \), find the range of values for \( a \).

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