
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 that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $30^{\circ}$, $|\overrightarrow{a}|=\sqrt{3}$, and $|\overrightarrow{b}|=2$.  
$(1)$ Find $\overrightarrow{a} \cdot \overrightarrow{b}$ and $|\overrightarrow{a}-\overrightarrow{b}|$;  
$(2)$ Find the angle $\theta$ between the vectors $\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{b}$.

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