
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 at the point $(1,f(1))$ that is perpendicular to $y=\frac{1}{2}x$

$(1)$ Find the value of $a$;

$(2)$ Determine the intervals of monotonicity and the extremum of the function $f(x)$.

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