
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 triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, $c\cos A= \frac{4}{b}$, and the area of $\triangle ABC$, $S \geq 2$.  
(1) Determine the range of possible values for angle $A$.  
(2) Find the maximum value of the function $f(x) = \cos^2 A + \sqrt{3}\sin^2\left(\frac{\pi}{2}+ \frac{A}{2}\right) - \frac{\sqrt{3}}{2}$.

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