
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 $\overrightarrow{a}=(1,\cos 2x)$, $\overrightarrow{b}=(\sin 2x,- \sqrt{3})$, and the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$.

$(1)$ If $f\left( \frac{\theta }{2}+\frac{2\pi }{3} \right)=\frac{6}{5}$, find the value of $\cos 2\theta$;

$(2)$ If $x\in \left[ 0,\frac{\pi }{2} \right]$, find the range of the function $f\left( x \right)$.

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