
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 $\overrightarrow{a}=(\sin x,\cos x),\overrightarrow{b}=(2\sqrt{3}\cos x-\sin x,\cos x)$, and $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.  
$(1)$ Find the interval where the function $f(x)$ is monotonically decreasing.  
$(2)$ If $f(x_0)=\frac{2\sqrt{3}}{3}$ and $x_0\in\left[\frac{\pi}{6},\frac{\pi}{2}\right]$, find the value of $\cos 2x_0$.

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