
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 $$f(x)=\cos 2x + 2\sqrt{3}\sin x\cos x$$  
(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing;  
(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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