
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 $\overrightarrow{a}=({2\cos x,\frac{{\sqrt{3}}}{2}})$ and $\overrightarrow{b}=({\sin(x-\frac{π}{3}),1})$, let $f(x)=\overrightarrow{a}⋅\overrightarrow{b}$.  
$(1)$ Find the values of $x$ when $f(x)$ reaches its maximum value.  
$(2)$ If ${x_0}∈[{\frac{{5π}}{{12}},\frac{{2π}}{3}}]$ and $f({{x_0}})=\frac{4}{5}$, find the value of $\tan({2{x_0}-\frac{π}{{12}}})$.

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