
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 the vector $\overrightarrow{a}=(\cos \theta,\sin \theta)$, where $\theta\in[0,\pi]$, and the vector $\overrightarrow{b}=(\sqrt{3},-1)$  
$(1)$ If $\overrightarrow{a}\perp \overrightarrow{b}$, find the value of $\theta$;  
$(2)$ If $|2 \overrightarrow{a}- \overrightarrow{b}| < m$ always holds, find the range of the real number $m$.

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