
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 $\overrightarrow{a}=({\frac{1}{2}\cos x,\frac{{\sqrt{3}}}{2}\sin x}),\overrightarrow{b}=({\sqrt{3},-1})$.<br/>$(1)$ When $\overrightarrow{a}⊥\overrightarrow{b}$, find the value of $\tan x$;<br/>$(2)$ Let $f(x)=({\overrightarrow{a}+\overrightarrow{b}})⋅\overrightarrow{b}$, and $x∈[{0,\frac{π}{2}}]$, find the maximum value of $f\left(x\right)$ and the corresponding value of $x$.

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