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

1. Find the maximum and minimum values of $f(x)$ in the interval $[0,\frac{\pi}{4}]$.
2. If $f(x_{0})=\frac{6}{5}$, $x_{0}\in[\frac{\pi}{4},\frac{\pi}{2}]$, find the value of $\cos 2x_{0}$.
3. If the function $y=f(\omega x)$ is monotonically increasing in the interval $(\frac{\pi}{3},\frac{2\pi}{3})$, find the range of positive values for $\omega$.

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