
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 $f(x) = \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2}$, where $a \in \mathbb{R}$, and the curve $y = f(x)$ has a tangent line at the point $(1, f(1))$ that is perpendicular to the line $y = \frac{1}{2}x$.

(1) Find the value of $a$;
(2) Determine the intervals where the function $f(x)$ is monotonic.

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