
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 $\overrightarrow{a}=(2\cos x,1)$ and $\overrightarrow{b}=( \sqrt {3}\sin x+\cos x,-1)$, let the function be $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.
$(1)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$;
$(2)$ If $f(x_0)= \frac{6}{5}$, where $x_0 \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the value of $\cos 2x_0$;
$(3)$ If the function $y=f(\omega x)$ is monotonically increasing on the interval $\left(\frac{\pi}{3}, \frac{2\pi}{3}\right)$, find the range of the positive number $\omega$.

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