
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 $f(x)=\cos 2x+2\sqrt{3}\sin x\cos x$, where $0\leqslant x\leqslant \frac{\pi }{2}$

(1) Find the maximum value of the function $f$ and specify the corresponding value of $x=0$.

(2) If $0 < \theta < \frac{\pi }{6}$ and $f(\theta )=\frac{4}{3}$, calculate the value of $\cos 2\theta$.

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