
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 vectors $\overrightarrow{a} = (1, \cos 2x)$ and $\overrightarrow{b} = (\sin 2x, -\sqrt{3})$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$.
(1) Find the interval of monotonic increase for the function $f(x)$;
(2) If $f\left( \frac{\theta}{2} + \frac{2\pi}{3}\right) = \frac{6}{5}$, find the value of $\cos 2\theta$.

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