
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 \(f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2 x - 1\),
(1) Find the maximum value of \(f(x)\) and the value of \(x\) when the function reaches this maximum value;
(2) In triangle \(ABC\), let \(a\), \(b\), and \(c\) be the sides opposite to angles \(A\), \(B\), and \(C\), respectively. If \(a= \sqrt{7}\), \(b= \sqrt{3}\), and \(f\left( \frac{A}{2}\right) = \sqrt{3}\), find the value of side \(c\).

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