
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 $\overrightarrow{a}=(2\cos x,1)$ and $\overrightarrow{b}=(-\cos (x+\frac{π}{3}),\frac{1}{2})$, where $x\in [0,\frac{π}{2}]$.  
$(1)$ If $\overrightarrow{a}∥\overrightarrow{b}$, find the value of $x$.  
$(2)$ Let $f(x)=\overrightarrow{a}•\overrightarrow{b}$. If for any $x_{1}$, $x_{2}\in [0,\frac{π}{2}]$, $|f(x_{1})-f(x_{2})|\leqslant \lambda$ always holds, find the minimum value of the real number $\lambda$.

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