
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 plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy: $\overrightarrow{a}=(\cos x, \sin x)$, $\overrightarrow{b}=(\sqrt{3}, -1)$.  
$(1)$ Find the maximum and minimum values of $|2\overrightarrow{a}-\overrightarrow{b}|$.  
$(2)$ Let $f(x)=\overrightarrow{a}\cdot(\overrightarrow{a}+\overrightarrow{b})$. If the equation of a symmetry axis of the graph of $f(x)$ is $x=x_{0}$, find the value of $f(x_{0})$.

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