
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 $\overrightarrow{a}=(1, \sqrt {3})$ and $\overrightarrow{b}=(-2,0)$:  
$(1)$ Find $|\overrightarrow{a}-\overrightarrow{b}|$;  
$(2)$ Find the angle between vector $\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}$;  
$(3)$ When $t\in\mathbb{R}$, find the range of $|\overrightarrow{a}-t\overrightarrow{b}|$.

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