
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 $\overrightarrow{m}=({\frac{1}{2},-\frac{{\sqrt{3}}}{2}})$ and $\overrightarrow{n}=({cosx,sinx})$.  
$(1)$ If $\overrightarrow{m}∥\overrightarrow{n}$, find the value of $\tan x$.  
$(2)$ If $\overrightarrow{m}⋅\overrightarrow{n}=\frac{1}{3}$ and $x∈({0,\frac{π}{2}})$, find the value of $\cos x$.

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