
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 $\overrightarrow{OA}=(\lambda \cos \alpha ,\lambda \sin \alpha )(\lambda \ne 0)$ and $\overrightarrow{OB}=(-\sin \beta ,\cos \beta )$, where $O$ is the origin of the coordinate system.

 $(1)$ If $\alpha -\beta =\frac{\pi }{6}$ and $\lambda < 0$, find the angle between vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$;

 $(2)$ If $|\overrightarrow{AB}|\geqslant 2|\overrightarrow{OB}|$ holds for any real numbers $\alpha ,\beta$, find the range of the real number $\lambda$.

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