
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 $f(x) = 4\ln x - ax + \frac{a+3}{x}$, where $a \geqslant 0$.
$(1)$ When $a = \frac{1}{2}$, find the extreme value of $f(x)$.
$(2)$ When $a \geqslant 1$, let $g(x) = 2e^{x} - 4x + 2a$. If there exist $x_{1}$ and $x_{2} \in [\frac{1}{2}, 2]$, such that $f(x_{1}) > g(x_{2})$, find the range of real number $a$. (where $e$ is the base of natural logarithm, $e = 2.71828\ldots$)

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